Poisson brackets of endomorphisms

Question

Let $(M,\omega)$ be a symplectic manifold and $(E,\nabla)$ a fiber bundle with connection. Let $W$ be a 2-form with endomorphisms. I define Poisson brackets of endomorphisms:

$$\{ e, f\}= W(\nabla e,\nabla f)$$

$$2 W( X^* \otimes e, \nabla f)= e. \nabla_X f+\nabla_X f .e$$

With $e,f$ two endomorphisms and $X$ a tangent vector.

Are the Poisson brackets well defined?