 Invertibility of identity plus a real skew-symmetric matrix

Proof. If $(I+A)x=0$, the quadratic-form criterion for a skew-symmetric matrix gives $0=x^{\top }(I+A)x=\Vert x{\Vert }^2$, so $x=0$. The rank-nullity theorem makes the square matrix $I+A$ invertible. More generally $tI+A$ is invertible for every nonzero real $t$, since its null space would imply $t\Vert x{\Vert }^2=0$. Equivalently, the only possible real eigenvalue of a real skew-symmetric matrix is zero.