In two dimensions, it is easy to explain why the determinant of a linear transformation works as a scaling factor of parallellograms, by showing that the formula is the same as the cross product.
Likewise, for three dimensions, the determinant could be shown to be a scaling factor by showing how it relates to the scalar triple vector product.
In 4 dimensions, I've read that the determinant still works as a scaling factor, though I have no idea why.
My problem with using the cross product as an explanation, is that the cross product "itself" is written as a determinant, so the proof just appears circular to me.
What I'm wondering, is if there is some deeper connection between the formula for the determinant(as a scaling factor) and the cross product, or if it's just a coincidence that they're the same.