[bcuccioli]

There are two good correct equivalent ways to think of the determinant of varying generality. One is as the function from a ring of square matrices to the underlying field (e.g. from R^(n^2) -> R) that sends identity to identity, is alternating (swapping two rows or columns negates the function) and is multilinear (is a linear function in each of the columns independently). These properties are all useful and important on their own, so there is motivation to study a function which has all of them. It's not obvious that such a function exists, but you can prove that. As it turns out, these three properties uniquely determine such a function, which makes it seem like that function might be really important!

There's a more general definition too, which is based around the wedge product, a quintessential object in algebra and calculus. There's a good exposition here: http://codeblank.com/~int/det.pdf .