This is probably a very elementary question. I'm trying to get an explicit description of the cochain complex and coboundary maps for Lie algebra cohomology over $\mathbb{Z}$, and more generally, over commutative unital rings that are not fields.

At first, I thought that the explicit descriptions given in these lecture notes would work:

http://www.scholarpedia.org/article/An_introduction_to_Lie_algebra_cohomology/lecture_2

However, it seems that, in their proof that the second cohomology classifies all the Lie algebra extensions, the authors use the fact that the "module extension" aspect of the extension splits. Hence, to specify a Lie algebra extension, they only need to describe the behavior of the Lie bracket. This works well over fields because any module over a field is free, but no longer works for Lie rings over the integers, particularly those with torsion.

Can somebody give either the explicit construction or a reference to an explicit construction of the cochain complex that works for $\mathbb{Z}$? I am aware that the complex can be defined using derived functors, but I am not sure how to translate that description into an explicit cochain complex analogous to the bar complex.