The determinant is uniquely determined by the following two axioms:
1. The determinant of the "multiply by lambda" operation on a one-dimensional vector space is lambda. 2. If you have a linear operator T on a vector space V, and a T-invariant subspace W, the determinant of T is the product of the determinant of T restricted to W and the determinant of the operator that T induces on the quotient space V / W.
This actually kinda intuitively meshes with the volume-stretching property: if you are stretching the volume by the factor lambda_1 along one subspace and by the factor lambda_2 along some complementary subspace, clearly the overall stretch factor is lambda_1 * lambda_2.
If you aren't working over an algebraically closed field, you can just tensor with the algebraic closure of whatever field you're working with and take the determinant there. There is also a way of adapting the definition so you don't have to do this, but it makes axiom (1) a bit more complicated.
Another bonus is this definition makes the Cayley-Hamilton theorem completely trivial.
Also, you can give an analogous definition of the trace if you replace multiplication with addition, and of the characteristic polynomial if you replace lambda in axiom 1 with x - lambda.