  We show the existence of families of orthonormal, future directed bases which
allow to cast every skew-symmetric endomorphism of $\mathbb{M}^{1,n}$
($\mathrm{SkewEnd}(\mathbb{M}^{1,n})$) in a single canonical form depending on
a minimal number of parameters. This canonical form is shared by every pair of
elements in $\mathrm{SkewEnd}(\mathbb{M}^{1,n})$ differing by an orthochronous
Lorentz transformation, i.e. it defines the orbits of the orthochronous Lorentz
group under the adjoint action on its algebra. Using this form, we obtain the
quotient topology of $\mathrm{SkewEnd}(\mathbb{M}^{1,n})/O^+(1,n)$. From known
relations between $\mathrm{SkewEnd}(\mathbb{M}^{1,n})$ and the conformal
Killing vector fields (CKVFs) of the sphere $\mathbb{S}^n$, a canonical form
for CKVFs follows immediately. This form is used to find adapted coordinates to
an arbitrary CKVF that covers all cases at the same time. We do the calculation
for even $n$ and obtain the case of odd $n$ as a consequence. Finally, we
employ the adapted coordinates to obtain a wide class of TT-tensors for $n=3$,
which provide Cauchy data at conformally flat null infinity $\mathscr{I}$.
Specifically, this class of data is characterized for generating
$\Lambda>0$-vacuum spacetimes with two-symmetries, one of which axial,
admitting a conformally flat $\mathscr{I}$. The class of data is infinite
dimensional, depending on two arbitrary functions of one variable as well as a
number of constants. Moreover, it contains the data for the Kerr-de Sitter
spacetime, which we explicitly identify within.
