Let $E$ be a Frechet space and $\mathcal{F}$ be a non-degenerate bounded skew symmetric bilinear map $\mathcal{F}: E\times E\to \mathbb R$ on $E$. We can identify $TE$ with $E\times E$, with this identification, $\mathcal{F}$ defines 2 form $\omega$ on $E$, by the following:
$$\omega\_x: T\_x E\times T\_xE\to \mathbb R$$
$$\omega\_x(v, w):= F(\tilde{v}, \tilde{w})$$
where $\tilde{v}$ and $\tilde{w}$ are identification of $v,w\in T\_x E$ with $E$.


I feel this definition is well defined for Frechet manifold and also the form defined in this way is closed. Is this true. 



Constant symplectic structure
Yes, it is. See Section 48 of


* [Peter W. Michor: Topics in Differential Geometry. Graduate Studies in Mathematics, Vol. 93 American Mathematical Society, Providence, 2008](http://www.mat.univie.ac.at/~michor/dgbook.pdf)


