We assume $n = 2m$ and denote by $J \in \mathcal { M } _ { 2m } ( \mathbb { R } )$ the matrix defined in blocks by $$J = \left( \begin{array} { c c }
0 & - I _ { m } \\
I _ { m } & 0
\end{array} \right)$$ and $j$ the endomorphism of $\mathbb { R } ^ { n }$ canonically associated with $J$. Show that the map $b _ { s } : \left\lvert \, \begin{array} { c c l } \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } & \rightarrow & \mathbb { R } \\ ( x , y ) & \mapsto & \langle x , j ( y ) \rangle \end{array} \right.$ is a symplectic form on $\mathbb { R } ^ { n }$.
We assume $n = 2m$ and denote by $J \in \mathcal { M } _ { 2m } ( \mathbb { R } )$ the matrix defined in blocks by
$$J = \left( \begin{array} { c c }
0 & - I _ { m } \\
I _ { m } & 0
\end{array} \right)$$
and $j$ the endomorphism of $\mathbb { R } ^ { n }$ canonically associated with $J$. Show that the map $b _ { s } : \left\lvert \, \begin{array} { c c l } \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } & \rightarrow & \mathbb { R } \\ ( x , y ) & \mapsto & \langle x , j ( y ) \rangle \end{array} \right.$ is a symplectic form on $\mathbb { R } ^ { n }$.