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.

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    $\begingroup$ Why do write "for even $N$"? If $A$ is always orthogonal, this also holds for uneven $N$. Or is an even $N$ the only case you are interested in? $\endgroup$ – Bobson Dugnutt May 21 '17 at 23:39
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    $\begingroup$ @Lovsovs: A non-singular skew-symmetric matrix must be of even dimension, since the only real eigenvalue it may have is $0$: if $A^T = -A$ and $Ax = \mu x$ with $x \ne 0$, then $\mu (x, x) = (x, Ax) = (A^Tx, x) = -(Ax, x) = -\mu (x, x)$, forcing $\mu = 0$. Since any real matrix of odd size has a real eigenvalue, $A$ must be of even size to be non-singular. $\endgroup$ – Robert Lewis May 22 '17 at 0:30

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