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Yes, I think $\Omega$ is closed. I will add a proof later. Yes, as a mapping $T\mathcal N \to T^*\mathcal N$ it is injective, but it can never be surjective, since $T_n\mathcal N$ is a Frechet space, whereas its dual $T^*_u\mathcal N$ is a DF-space (generalized functions of distributions) which can never be isomorphic to a Frechet space. So $\Omega$ is a weak symplectic structure. See section 48 (called: Weak Symplectic Manifolds) of here, where 48.2 and 48.8 have to be corrected as described in the Errata.