In da Silva's Lectures in Symplectic Geometry page 105, it was claimed that if $\omega$ is nondegenerate symplectic 2-form then there exists unique vector field $X_H$ such that $\iota_{X_H}\omega=dH$ for some smooth function $H$.
I cannot quite see this intuitively. Naively, this seems to be a fair claim since at least $d^2H=0$ showing that indeed $\omega$ is closed. But I cannot see the principle behind it: is this a statement of Poincar\'e lemma? That this is true is somewhat remarkable to me, unless I am misunderstanding differential forms in general to miss this obvious statement (I think it is not obvious).
Non-degeneracy means the map
$$\omega^{\sharp} : TM \to T^* M$$ $$\omega^{\sharp}(X)(Y) = \omega(X, Y)$$
is an isomorphism (hence it's also an isomorphism on sections). So you just need to set $X_H = (\omega^{\sharp})^{-1}(dH)$.