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?

Let $B$ a basis of a vector space $E$ of dimension $n$ over $\Bbbk$. Then $det_B$ is the only $n$-alternating multilinear form with $det_B(B) = 1$.
A $n$-multilinear form is a map of $E^n$ in $\Bbbk$ which is linear for each variable.
A $n$- alternated multilinear form is a multilinear form which verify for all $i,j$
$$ f(x_1,x_2,\dots,x_i,\dots, x_j, \dots, x_n) = -f(x_1,x_2,\dots,x_j,\dots, x_i, \dots, x_n) $$
In plain english, the sign of the application change when you switch two argument. You understand why you use the big sum over permutations to define the determinant with a closed formula. 
