The Morrey--Kohn--Hormander identity is the key to proving vanishing/existence results on bounded pseudoconvex domains in $\mathbb{C}^n$, or more generally, Stein domains. See, for instance, the classical references Kohn and Hörmander, or the more recent exposition of Straube.
There is no lack of derivations of this identity in the literature, yet my question is still simply a "reference request" for a proof of this identity which doesn't use "calculation in coordinates" as the main tool in the proof. At first glance, it seems such a proof should not be that hard, since the proof is "just integration by parts", but on the other hand going from a calculation in coordinates to the corresponding coordinate free calculation is often nontrivial.
I have almost worked out a coordinate free proof of the identity (it remains "just" to put the boundary terms in the correct form). I hope, though, that such a proof has already been written, so I can cite it and check that what I've done so far is correct. In particular, I'm looking for a proof which gives a coordinate free calculation of the boundary term in the identity (I already know how to derive the interior terms from the Bochner--Kodaira--Nakano identity and some Kähler identities).