Show that if $M \in M _ { n } ( \mathbb { R } )$ is antisymmetric, the matrix $$O = \left( I _ { n } + M \right) ^ { - 1 } \left( I _ { n } - M \right)$$ is orthogonal.
Show that if $M \in M _ { n } ( \mathbb { R } )$ is antisymmetric, the matrix
$$O = \left( I _ { n } + M \right) ^ { - 1 } \left( I _ { n } - M \right)$$
is orthogonal.