Let $\nabla$ be a Levi-Civita covariant derivative on a riemannian manifold $(M,g)$ and let $\nabla^E$ be a covariant derivative on a vector bundle $E$ over $M$. We can understand $\nabla^E$ as a map
\begin{equation} \nabla^E:\Gamma(E)\to\Gamma(T^\ast M\otimes E) \end{equation}
between section spaces, so that we can define recursively
\begin{equation} (\nabla^E)^k:\Gamma(E)\to\Gamma(\bigotimes^k T^\ast M\otimes E)\ . \end{equation}
It is known that the formal adjoint of $\nabla$ (with respect to the riemannian metric $g$ and some hermitean structure on $E$) is given by
\begin{equation} \nabla^\ast_X=-div(X)-\nabla_X \end{equation}
for each vector field $X$ on $M$, and also that, in general, if $P:\Gamma(E)\to\Gamma(F)$ is linear differential operator (here $F$ is another hermitean vector bundle over $M$), then $P$ has a (unique) formal adjoint $P^\ast$ if there exists a $\omega\in\Gamma(E^\ast\otimes F^\ast\otimes\bigwedge^{k-1}M)$ such that
\begin{equation} \langle P\alpha,\beta\rangle_E-\langle\alpha,P^\ast\beta\rangle_F=d(\omega(\alpha,\beta))\ , \end{equation}
for every compactly supported pair of sections $\alpha,\beta$, essentially by Stokes' theorem.
My question is: is there a general extension of the formula above for $\nabla^\ast$ for the formal adjoint of the $k$-th order covariant derivative $\nabla^k$?