A symplectic flow

Question

Let $(M,g)$ be a riemannian manifold. I consider a flow over symplectic forms $\omega$. I define a connection $\nabla$ bounded with $g$ and $\omega$. The symmetric part of $\nabla$ is defined by the fact that $\nabla$ conserves $g$ and the anti-symmetric part of $\nabla$, by the fact that $\omega$ is conserved by $\nabla$. The torsion of $\nabla$ is $T_{\nabla}$. Then I define a flow over $\omega$ by the equation:

$$\frac{\partial \omega}{\partial t}(X,Y)=\omega (T_{\nabla}(e_i, X),T_{\nabla}(e_i,Y))$$

Have we solutions of the symplectic flow for short time?