Let $(M^{2n},\omega)$ be a closed connected symplectic manifold and let 
$HR : H^2(M;R) \times H^2(M;R) \rightarrow R$ be the Hodge-Riemann form defined by
$HR(\alpha,\beta) = \int_M \alpha \beta \omega^{n-2}$. 

I wonder when $HR$ is non-singular.
We can easily show that $HR$ is non-singular if and only if $\omega^{n-2} : H^2(M;R) \rightarrow H^{2n-2}(M;R)$ is an isomorphism.
Of course if $\omega$ is Kaehler or of Hard Lefschetz type, then it is true. 

My question is, is there any other condition that makes $HR$ to be non-singular?

And if you know the examples such that $\omega^{n-2} : H^2(M;R) \rightarrow H^{2n-2}(M;R)$
is not an isomorphism, please let me know. 

Thank you in advance.