  For each quadratic form $Q\in \mbox{Quad}(V)$ over a given vector space over
a field $\mathbb{R}$ we have the Clifford algebra $Cl(V,Q)$ defined as the
quotient $T(V)/I(Q)$ of the tensor algebra $T(V)$ over the two-sided ideal
generated by expressions of the form $x\otimes x-Q(x),\, x\in V.$ In the
present paper we consider the whole family $Cl(V,Q),\, Q\in \mbox{Quad}(V)$ in
a geometric way as a $Z_2$-graded vector bundle over the base manifold
$\mbox{Quad}(V).$ Bilinear forms $F\in\mbox{Bil}(V)$ act on this bundle
providing natural bijective linear mappings $\bar{\lambda}_F$ between Clifford
algebras for different $Cl(V,Q).$ Alternate (or antisymmetric) forms induce
vertical automorphisms, which we propose to consider as 'gauge
transformations'. We develop here the formalism of N. Bourbaki, which
generalizes the well known Chevalley's isomorphism
$Cl(V,Q)\rightarrow\mbox{End}(\bigwedge(V))\rightarrow\bigwedge(V).$ In
particular we realize the Clifford algebra twisting gauge trnsformations
induced by antisymmetric bilinear forms as exponentials of contractions with
elements of $\bigwedge(V^*)$ representing these forms. Throughtout all this
paper we intentionally avoid using the so far accepted term "Clifford algebra
of a bilinear form" (known otherwise as "Quantum Clifford algebra"), which we
consider as possibly misleading, as it does not represent any well defined
mathematical object. Instead we show explicitly how any given Clifford algebra
$Cl(Q)$ can be naturally realized as acting via endomorphisms of any other
Clifford algebra $Cl(Q')$ if $Q'=Q+Q_F,\, F\in \mbox{Bil}(V)$ and
$Q_F(x)=F(x,x).$ Possible physical meaning of such transformations are also
mentioned.
