A definition of a connection

Question

Let $(M,g,\omega)$ be a riemmannian manifold with metric $g$ and a $2$-form $\omega$. I define a connection $\nabla$ by the following equations:

$$X.g(Y,Z)=g(\nabla_X Y,Z)+g(Y,\nabla_X Z)$$

$$X.\omega(Y,Z)=\omega (\nabla_X Y,Z)+\omega (Y,\nabla_X Z)$$

Is the connection $\nabla$ well defined and unique?