Except they can be negative, and relate to area for 2D, not volume, and would require defining volume in dimensions higher than 3, which is not necessarily obvious or intuitive for beginners.

"Signed scaling factor" is more general, and some courses (e.g. ones I teach on) use that to introduce the idea.

I think that 'N-volume' including area, volume, and higher dimensional analogs is pretty understandable to a student. The fact that it's signed is more subtle, but still is not a big problem. The benefits to intuition of calling it 'volume' far outweigh the negatives.

Anyway, 'scaling factor' still leaves the signs mysterious.

Volume of what?

Of the parallelepiped generated by the columns of the matrix. (https://textbooks.math.gatech.edu/ila/determinants-volumes.h...)

The result of the determinant might be negative, which doesn't make sense for a volume, so you need to take the absolute value to interpret it that way.

Another way to think of it is the change in volume caused by the linear transformation determined by the matrix (http://algebra.math.ust.hk/determinant/01_geometry/lecture3....)