I found the following reasoning that, it seems immediate in the document, but for me it is not. It comes from tangent spaces of codimension 1 submanifolds on a symplectic manifold, but it can be stated in terms of symplectic vector spaces in the following way.
If $W$ is a codimension 1 subspace of a symplectic vector space $(V,\Omega)$, then the $\Omega$-complement $W^\Omega \subseteq W$.
I can notice that $\Omega\vert_{W \times W}$ is degenerated since $W$ is odd dimensional, so there should be a non-zero vector $w \in W$ such that $\Omega(w, v) = 0$ for all $v \in W$, but I do not how to see that $W^\Omega$ is actually spanned by that vector $w$.