We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ with associated matrix $\Omega = \left( \omega \left( e _ { i } , e _ { j } \right) \right) _ { 1 \leqslant i , j \leqslant n }$ satisfying $\omega(x,y) = X^{\top} \Omega Y$ for all $x,y \in \mathbb{R}^n$. Deduce that $\Omega$ is antisymmetric and invertible.