If we're talking about the Lie algebra of space rotations, then the elements of that are infinitesimal rotations. Those elements form a vector space, and there's also the Lie bracket, which is the cross product as you say.

I don't see the sense in saying "rotate b about a" when both a and b are to represent infinitesimal rotations. If we're taking b to be a position in space, and a to be an infinitesimal rotation, then your explanation makes sense (though in a backwards way from how I think of it. a x b is the vector you rotate a about in order to get the derivative of the tip to be b). How do you reconcile that with the Lie Algebra?

More importantly, what if your vectors don't model rotations to begin with? Then you still are left with giving a meaning to the cross product. How can you clarify anything by taking your vectors to be infinitesimal rotations, whose cross product gives a third infinitesimal rotation, which carries information about the non-commutativity of the [non-infinitesimal] rotations?

It is one thing that the cross product does, but I don't see how it explains, for example, what the cross product has to do with calculating volumes of parallelepipeds, or torques.