I want to find a surface and a smooth 2-form on it that is not of the form $d\omega$.
I know a theorem that says: every closed 1-form on $S^2$ is exact. So my hunch is that $S^2$ is a candidate for a surface. It isn't clear to me how to really construct such a 2-form though.
On any orientable compact manifold, the volume form $\omega$ is not exact. If it were, then it would integrate to zero by Stokes' theorem, which contradicts the fact that it is positive everywhere.
So you can take $M = S^2$ and $\omega = \sin \theta \, \mathrm{d}\theta \wedge \mathrm{d} \phi $.