Let $k$ be a field whose characteristic is zero and let $n\geq 1$. Say that a matrix $M\in {\cal M}_{n\times n}(k)$ is almost orthogonal if $M^{T}M$ is a nonzero multiple of the identity. Denote the set of those matrices by $AO_n(k)$.
When $n=2$, there is a nice parametric description of $AO_2(k)$ :
$ AO_2(k)=\Bigg\lbrace \bigg(\begin{matrix} a & b \\ -b&a \end{matrix}\bigg) \Bigg| (a,b) \in k^2, (a,b)\neq (0,0) \Bigg\rbrace $
Are there similar exhaustive formulas with polynomial entries for $n>2$ ?