We fix a symplectic form $\omega$ on $E$. Show by induction that there exists a basis $\widetilde { \mathcal { B } }$ of $E$ such that
$$\operatorname { Mat } _ { \widetilde { \mathcal { B } } } ( \omega ) = \left( \begin{array} { c c c c } J _ { 2 } & 0 & \cdots & 0 \\ 0 & J _ { 2 } & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & J _ { 2 } \end{array} \right)$$