Unique Alternating Multilinear D:R^n->R

Edwards writes: "There exists a unique (that is, one and only one)
multilinear alternating function D, from n-tuples of vectors in R^n to real
numbers, such that D(e_1,...,e_n)=1." He also indicates that it is beneath
him to explain such trivialities, and that I should consult the chapter on
determinants in my Linear Algebra textbook. Well, that is not how the
Linear Algebra textbooks (I emphasize the plural) I consulted treat the
determinant. The definition for the determinant which I find easiest to
remember is the contraction of the Levi-Civita symbol with the elements of
the matrix appearing once per dimension, and putting the free indices in
ascending order.

I haven't given a lot of thought to showing these definitions are
equivalent. It may be fairly easy. Nonetheless, Edwards's definition
suggests an approach worth investigating further. Starting with his
definition certainly makes the subsequent proofs trivial. Crammer's rule
is almost embarrassingly simple in that form. Is this a fairly common
means of defining the determinant? Is there a good and concise treatment
of this approach available?

IIRC the last time I asked about this someone pointed me to a source on
Grassmann. I am very interested in pursuing that, but I don't believe it
would be wise in the immediate future to do so.
--
Nil conscire sibi
.