If we have a covariant derivative for vector fields given by an affine connection on a manifold, can we extend that to a covariant derivative for k-vectors by assuming that the product rule hold for wedge products of vectors? In other words, does it make sense to assume that:
$$\nabla_v(a\wedge b) = (\nabla_va)\wedge b+a\wedge (\nabla_vb) $$
where $v$ is a vector field, $a$ and $b$ are $p$-vector and $q$-vector fields, and $\nabla_va$ agrees with the covariant derivative of vectors when $a$ is a vector field?
(I am wondering since the product rule for the exterior derivative has a grade-dependent sign in it, but I think that should not be the case here.)