Tricky problem on skew-symmetric matrices
I will assume we are in $M_n(\mathbb{R})$, since you want your matrix to be orthogonal.
The spectrum of a skew-symmetric matrix is contained in $i\mathbb{R}$, so the spectrum of $A=I-S$ does not contain $0$ and $A$ is invertible, in $M_n(\mathbb{C})$ first, hence in $M_n(\mathbb{R})$.
The matrix we are interested in is $A^*A^{-1}$. Note that $A$ and $A^*$ commute.
Now $$ A^*A^{-1}(A^*A^{-1})^*=A^*AA^{-1}(A^{-1})^*=A^*AA^{-1}(A^*)^{-1}=A^*A(A^*A)^{-1}=I. $$
So $A^*A^{-1}$ is indeed orthogonal.