Skew symmetric subtracted from Identity

Question

How can I prove that for any skew-symmetric matrix $S$ with $S^T = -S$, $I - S$ is non-singular and $(I-S)^{-1}(I+S)$ is an orthogonal Matrix (cayley transform of S).

Answer 1

Let $Sx=x $,then $S^TSx=S^Tx=-Sx=-x $. Since $S^TS $ is positive semidefinite, $-1$ is not an eigenvalue of $S^TS $, hence $x=0$.