Universal parametrization for orthogonal matrices 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 $n>2$ ? 
 A: Let's talk about $SO(n)$ instead. You do this as well implicitely, since the determinant of the matrix you have written is $a^2 + b^2 \geq 0$, so that this obviously can't parametrize even $O(2)$ which also has a component with negative determinant. The full parametrization would be two copies of $AO_2(k)$ glued along the surface of zero determinant (if you allow the multiple of the identity to be zero in your definition).
Now, this is purely a consequence of low dimension. $SO(2)$ is a circle and if you allow $M^TM$ be a multiple of identity then you also allow each matrix from $SO(2)$ to be multiplied by any number, so that you obtain a line bundle over $SO(2)$ surface and this is more or less just polar parametrization of the plane that the $SO(2)$ lies in.
In higher dimensions, you similarly obtain a line bundle over $SO(n)$. But since $SO(n)$, $n>2$ is a much more complicated surface (as a subspace of the space of all matrices) than a simple circle, you can't expect anything nice here for big $n$.
