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That paper looks extremely applicable, but it is a little hard to approach. Is there any way you can help distill their result? In the Cayley Transform you parameterize with a skew-symmetric matrix and the function that produces the square orthogonal matrix is straightforward - but I'm not sure what that function looks like from this paper :(
I'm so sorry - I had the sign backwards on the $N = 1$ case, but your answers still works (just now without the negative). In the $N > 1$ cases, I think the same logic still works, only $C = (W_0W_1 \dots W_N)^T$. But I am hoping for some wider class of $C$ that works (since the $C=W^T$ constraint is pretty limiting)
