<PAGE>   1
                                                                     EXHIBIT 4.4


                                                                            1998



         THIS INSTRUMENT IS SUBORDINATED TO THE EXTENT AND IN THE MANNER
         PROVIDED IN THE SUBORDINATION AGREEMENT REFERRED TO BELOW.

         NOTWITHSTANDING THE PROVISIONS OF SECTION 2.05 OF THE SECURITIES
         PURCHASE AGREEMENT REFERRED TO BELOW, THE INITIAL PRINCIPAL BALANCE OF
         THIS NOTE IS $____________, AND SUCH PRINCIPAL AMOUNT SHALL BE
         INCREASED BY THE AMOUNT OF ANY PAYMENTS OF INTEREST ON THIS NOTE IN
         KIND.  FOR MORE INFORMATION, CONTACT THE COMPANY AT THE ADDRESS
         SPECIFIED IN THE SECURITIES PURCHASE AGREEMENT.

                                ________________

                   Senior Subordinated Secured Note due 2003

No. __                                                            Houston, Texas
$_________                                                       August __, 1998

         BRIGHAM EXPLORATION COMPANY, a Delaware corporation ("Company"), for
value received, hereby promises to pay to the order of ___________
("Purchaser"), to its account at __________________, ABA No. _________, Account
No. __________, or such other account or address in the State of Texas as may
be specified by the Purchaser, the principal amount of ___________ DOLLARS
($________), or, if less, the aggregate unpaid principal balance of this Note
(including any additions to such principal balance as a result of the payment
of interest owing in respect of this Note in kind, as more fully described
below), together with interest on the unpaid balance of such principal amount
in accordance with the Indenture dated as of August __, 1998 (the "Indenture"),
between the Company, and ________________, as Trustee (the "Trustee").  The
Indenture is being executed in connection with the Securities Purchase
Agreement dated as of August __, 1998 (as modified from time to time, the
"Securities Purchase Agreement", and together with the Indenture, the
"Securities Purchase Documents") among the Company, the Purchaser and the other
purchasers party thereto (together with the Purchaser, referred to collectively
as the "Noteholders"), and ______________________________, as agent for the
Noteholders ("Agent").
<PAGE>   2
         The Securities Purchase Documents provide for the making of loans by
the Noteholders to the Company in exchange for, among other things, the
issuance by the Company of the Notes and Warrants (as defined in the Securities
Purchase Documents).  The Company hereby acknowledges and agrees that, as of
the date of the issuance of this Note and notwithstanding the provisions of
Section 2.05 of the Securities Purchase Agreement, the aggregate outstanding
principal balance of this Note is $_________. In addition, the Securities
Purchase Documents provide that payment of a portion of the interest owing by
the Company in respect of the Notes may be made by the Company in kind.  The
Company hereby acknowledges and agrees that any such payments of interest in
kind in respect of this Note shall be deemed to be additional principal
advanced by the Purchaser under this Note and such additional principal shall
be added to the unpaid principal amount of this Note owing by the Company to
Purchaser in respect of this Note.  The loan made by the Purchaser to the
Company under this Note (including, without limitation, loans deemed to be made
as a result of any payments of interest in kind) and all payments made on
account of the principal amount of such loans shall be entered by the Purchaser
in its records or on the grid attached hereto which is part of this Note;
provided that, the failure of the Purchaser to make any such notation shall not
impair or otherwise affect the Company's obligations under this Note.

         This Note is subject to optional prepayments, in whole or in part and
without premium or penalty, as specified in the Securities Purchase Documents.
Payments of principal and interest on this Note are subordinated to the extent
provided in the Subordination Agreement dated as of August __, 1998
("Subordination Agreement") among the Noteholders, the Agent and Bank of
Montreal, as agent for certain senior bank lenders.  Payments of principal and
interest due on this Note shall be made in lawful money of the United States of
America in accordance with the Securities Purchase Documents.

         This Note is issued pursuant to the Securities Purchase Documents and
is subject to and entitled to the benefits of the Securities Purchase Documents
and the security and support therefor, and the holder of this Note may enforce
such rights in accordance with the Securities Purchase Documents.  Without
limiting the foregoing, upon the occurrence of an Event of Default (as defined
in the Indenture), payments due under this Note may be accelerated in the
manner and with the effect provided in the Securities Purchase Documents.

         It is the intention of the Purchaser and the Company to conform
strictly to any applicable usury laws.  Accordingly, the terms of the
Securities Purchase Documents relating to the prevention of usury will be
strictly followed.

         THIS NOTE SHALL BE GOVERNED BY AND CONSTRUED AND ENFORCED IN
ACCORDANCE WITH THE LAWS OF THE STATE OF TEXAS.



                                     -2-
<PAGE>   3


                                        BRIGHAM EXPLORATION COMPANY



                                        By:
                                           ------------------------------
                                        Name: Craig M. Fleming
                                        Title: Chief Financial Officer

TRUSTEE'S CERTIFICATE
OF AUTHENTICATION

This is one of the securities referred to
in the within-mentioned Indenture

CHASE BANK OF TEXAS, NATIONAL ASSOCIATION


By: 
   ------------------------------
       Authorized Signatory

Dated:
      ------------------



                                      -3-
<PAGE>   4
LOAN, MATURITIES, AND PAYMENTS OF PRINCIPAL



<TABLE>
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Date     Amount of   Maturity of      Amount of        Unpaid Principal    Notation Made
           Loan         Loan        Principal Paid          Balance               By
                                     or Prepaid
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<S>      <C>         <C>            <C>                <C>                 <C>

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</TABLE>



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