Let $f$ be an antisymmetric endomorphism of $E$. Show that, if $S$ is a vector subspace of $E$ stable under $f$, then $S^{\perp}$ is stable under $f$. Show that the endomorphisms induced by $f$ on $S$ and on $S^{\perp}$ are antisymmetric.
Let $f$ be an antisymmetric endomorphism of $E$. Show that, if $S$ is a vector subspace of $E$ stable under $f$, then $S^{\perp}$ is stable under $f$. Show that the endomorphisms induced by $f$ on $S$ and on $S^{\perp}$ are antisymmetric.