Spinorial symplectic form

Question

Let $(M,g)$ be a spin manifold, the vector bundle of spinors admit a hermitian product $<,>$. I define a 2-form on $M$ by the formula:

$$\omega (X,Y)= Im(<X.\psi, Y .\psi>)$$

where $\psi$ is a spinor, I take the imaginary part of the hermitian product.

What is the condition on the spinor $\psi$ for the 2-form $\omega$ to be a symplectic form?