The expression for combining two dice is pretty simple, isn't it?

Let's say that d_2 is the value that the d2 gives you (in {1,2}). Then to get d_6:

(d_2 - 1)*3 + d_3

or, in general,

(d_i - 1)*j + d_j

It's just a very slight generalization of what you do to use two d10s to give you a d100.

I think the further generalization to n dice should be straightforward. If I had a couple of minutes, I'd work it out now.