We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ and we denote by $\Omega \in \mathcal { M } _ { n } ( \mathbb { R } )$ the matrix defined by $$\Omega = \left( \omega \left( e _ { i } , e _ { j } \right) \right) _ { 1 \leqslant i , j \leqslant n }$$ where $(e _ { 1 } , \ldots , e _ { n })$ denotes the canonical basis of $\mathbb { R } ^ { n }$. Show that $$\forall ( x , y ) \in \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } , \quad \omega ( x , y ) = X ^ { \top } \Omega Y$$ where $X$ and $Y$ denote the columns of the coordinates of $x$ and $y$ in the canonical basis of $\mathbb { R } ^ { n }$.
We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ and we denote by $\Omega \in \mathcal { M } _ { n } ( \mathbb { R } )$ the matrix defined by
$$\Omega = \left( \omega \left( e _ { i } , e _ { j } \right) \right) _ { 1 \leqslant i , j \leqslant n }$$
where $(e _ { 1 } , \ldots , e _ { n })$ denotes the canonical basis of $\mathbb { R } ^ { n }$. Show that
$$\forall ( x , y ) \in \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } , \quad \omega ( x , y ) = X ^ { \top } \Omega Y$$
where $X$ and $Y$ denote the columns of the coordinates of $x$ and $y$ in the canonical basis of $\mathbb { R } ^ { n }$.