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it is a well-known result that for any skew-symmetric matrix $A$, that is invertible, we have invertible matrices $R$, such that $R^T A R = \left(\begin{array} 00 & Id \\ -Id & 0 \end{array} \right)$. Now since this looks extremely similar to what I know sympletic forms, I was wondering how symplectic forms are related to skew symmetric matrices. Can we even say, that every skew-symmetric matrix that is invertible "creates" a symplectic forms?

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For every invertible skew symmetric matrix $A$, the bilinear form $(x,y)\mapsto x^t\cdot A\cdot y$ is a symplectic form.

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    $\begingroup$ The converse is also true. $\endgroup$ – Michael Albanese Sep 3 '13 at 11:49

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