This year I'm teaching an elementary course on linear algebra for physics students. Because of that I have been researching the different ways of presenting the definition of the determinant (one of the hardest topics for such a course, in my opinion).

In the lecture notes by Terry Loring a definition using elementary row operations is given 

<http://www.math.unm.edu/~loring/links/linear_s06/det_def.pdf>

I like this definition since seems to me very algorithmic, and uses
an idea which is familiar to the students. Compare this with the other
more popular definitions which are

* The explicit formula using the sign of permutations
* The recursive formula using the Laplace development
* The axiomatic one as the unique multilinear alternate form by rows
(or by columns) that takes the value 1 at the identity matrix.

Of course, this definition by row elimination seems very close to this last axiomatic definition.

However my question is, do you know if there is some way of deriving the
whole theory of determinants from the definition based on row elimination?
Indeed, it even seems hard to prove directly that the definition is correct (non ambiguous). 

Do you know some book using this approach?