I've read in many places, including the n-Lab page, that a Lie algebroid (which I think of as in the first definition on the n-Lab page) is the same as a vector bundle $A \to X$ and a (properties?) derivation that makes $\Gamma(\wedge^\bullet A^*)$ into a cochain complex (here $A^* \to X$ is the dual vector bundle). One motivation for considering this cochain complex is that in the two extremes $A = {\rm T}X$ and $X = \{\rm pt\}$, the complex becomes respectively the deRham complex and the Chevellay-Eilenberg complex for a Lie algebra. My motivation has something to do with understanding BRST-type arguments.

I only partially understand this relationship. Given a trivialized vector bundle $(V\times X)\to X$ with a differential $d$ on $\mathcal C^\infty (X,\wedge^\bullet V)$, then I can define a Lie algebroid structure on $V\times X \to X$ — the Lie algebroid axioms are equivalent to $d^2 = 0$. Conversely, given a Lie algebroid $A\to X$ with $A$ trivialized as a bundle, I know how to interpret the standard formulas for the differential on $\Gamma(\wedge^\bullet A^*)$ — n-Lab, for example, reproduces the standard formula.

But I do not know how to interpret the formula defining the differential without trivializing the bundle, and my interpretation, as far as I can tell, depends on the trivialization (the fiber coordinates). At least, I've tried to check that my interpretation of the formula (which does get the above statements about $d^2 = 0$, etc., correct) is invariant under changing the trivialization, and I have failed. Hence:

What is a totally invariant description of the Lie algebroid CE chain complex?

Better: is there such a description that is simultaneously useful for explicit calculations?