An orientable manifold which does not admit a non degenerate $2$- form

Question

What is an example of an orientable manifold of dimension greater than $1$ such that it does mot admit a nondegenerate $2$-form?

Answer 1

Existence of a non-degenerate $2$-form requires the dimension to be even. Thus $\mathbb R^3$ is an example. If $\omega$ is a non-degenerate $2$-form on a manifold $M$ of dimension $2n$, then $n$-fold wedge product of $\omega$ is a volume form. Hence $M$ is orientable. Therefore there is no non-degenerate $2$-form on the Mobius strip or on the real projective plane. If $M$ is orientable and compact, then the cohomology class of $\omega^n$ is nonzero. Therefore the same is true for $\omega$, so the second Betti number of $M$ is nonzero. In particular the only sphere admitting a non-degenerate $2$-form is $S^2$.

Note It was pointed out in the comments that the part about the second cohomology class is false, as I have implicitly assumed that the form under question is closed, hence symplectic.