A non-degenerate 2-form $ \omega $ on a manifold $X$ is called an almost symplectic form. A manifold possesses such a form if and only if it is almost complex. One doesn't necessarily have a closed symplectic form however, for example, the $6$-sphere $S^6$ can't be symplectic as $ H^2 (S^6) =0$.
Suppose that $X$ is a closed almost symplectic manifold which doesn't possess a symplectic form. Can one always find an almost symplectic form $ \omega$ that is $C^0$-away from $0$, while the exterior derivative $ d \omega $ is arbitrarily $C^0$-close to $0$? For instance, can we fix a Riemannian metric $g$ and then find a family of almost complex structures $J_t$ so that the associated almost symplectic forms $ \omega_t $ satisfy $ d \omega_t \rightarrow 0$ as $ t \rightarrow \infty$?
Edit: Instead of $ \omega$ being away from $0$, one should fix a volume form $ \mu$ and constrain $ \omega^n = \mu$. This seems more meaningful since one can kill symplecticity while being away from $0$. I really wonder if the question is well posed enough. I'm fearful that it just sounds okay but is essentially meaningless.
