Suppose we have a real symplectic manifold $(M,\omega)$. Under what conditions can we find a global 1form $\alpha$ such that $\omega = \alpha \wedge\alpha$?
Obviously there are some simple obstructions, for example, the cotangent bundle must admit a nonvanishing section (thus, surfaces of genus $>1$ are out). However it does not seem easy to come up with sufficient conditions.



You can never find such a $1$form since any $1$form wedge itself is identically zero. 

