Let $(M,\omega)$ be a symplectic manifold. I'm trying to see why the non-degeneracy of $\omega$ is equivalent to say that the contraction map $X \longmapsto i_{X}\omega$ defines an isomorphism between vector fields on $M$ and differential $1$-forms on $M$.
So clearly the condition of non-degeneracy means injectivity, but what about surjectivity? How do we show that this mapping is surjective?
This is a pointwise issue. You have the map $X\mapsto \omega(x)(X,\_)$ which defines a linear map $T_xM\to (T _xM)^*$ for each point $x$. And since these are finite dimensional vector spaces, injectivity is equivalent to surjectivity.