By Cartan's magic formula and using that $\omega$ is by definition a closed form, the equation $\mathcal{L}_v \omega = 0$ is equivalent to

$d_{dR} \iota_v \omega = 0
\,,$

hence equivalent to the condition that the contraction of $v$ in $\omega$ is a closed differential form?. If this contraction even is an exact differential form? in that there is a function $h \in C^\infty(X)$ such that