Let $\omega \in \Omega^2(M)$ be a non-degenerate 2-form. For a function $f \in C^\infty(M)$, define the vector field $X_f ∈ \mathscr X(M)$ via the relation: $$i_{X_f} \omega = df$$, where the first member is the interior multiplication and define the following binary operation on $ C^\infty(M)$ $$\{f, g\} := X_f (g)$$.
Show that
$dω = 0 \iff$ {·, ·} is a Poisson bracket.
I am new to the concept of interior multiplication and not very comfortable manipulating differentials, so I have no idea how to go about solving this. Any help is appreciated