I'm currently trying to parameterize a given real and square matrix $A$ with the properties $A^T=-A$ and $A^TA=\textbf{1}_N$, for even $N$. I don't know how many independent parameters I would have for a given choice of $N$ and I would like to have a formula and a proof for this, but I don't know how. For instance, a general orthogonal matrix has $N(N-1)/2$ independent parameters.
I've found that if $A$ is a $4 \times 4$, then I have $2$ independent parameters, if there are no further restrictions.