Is there a definition of determinants that does not rely on how they are calculated? In the few linear algebra texts I have read, the determinant is introduced in the following manner;
"Here is a formula for what we call $detA$. Here are some other formulas. And finally, here are some nice properties of the determinant."
For example, in very elementary textbooks it is introduced by giving the co-factor expansion formula.  In Axler's "Linear Algebra Done Right" it is defined, for $T\in L(V)$ to be $(-1)^{dimV}$ times the constant term in the characteristic polynomial of $T$.
However I find this somewhat unsatisfactory. Its like the real definition of the determinant is hidden. Ideally, wouldn't the determinant be defined in the following manner:
"Given a matrix $A$, let $detA$ be an element of $\mathbb{F}$ such that x, y and z."
Then one would proceed to prove that this element is unique, and derive the familiar formulae.
So my question is: Does a definition of the latter type exist, is there some minimal set of properties sufficient to define what a determinant is? If not, can you explain why?

In Artin's book, Algebra, the determinant is defined as the only function $\det: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}$ such that 
1 - $\det(I) = 1$ 
2 - $\det$ is an $n$-multilinear function
3 - if two rows (columns) of $A$ are equal, then $\det(A) = 0$
Which, I think, can be proven using Qiaochu's answer (but I'm not sure).
