Well, here is the most general possible definition of "determinant". Given an n by n matrix, construct all possible products taking exactly one number from each row and column. It is easy to see that there are n! such products- you can choose any of the n numbers in the first column as the first number, then choose any number in the second column except the one in the row you chose before; n- 1 choices; then any of the numbers in the thirde column except the ones in those two previously chosen rows; n- 2 choices; etc.
Each such product can be written ordered by column: . Because there is exactly one number from each column, the " " of second indices is a permutation of "1 2 3... n". Multiply each product by 1 if that is and even permutation and by -1 if it is an odd permutation and add.
(Fortunately, there are much better ways of actually calculating the determinant.)
Those positive and negative terms will cancel, giving a determinant of 0 if and only if the odd and even permutations contain the same " " which is the same as saying one row or column is a multiplie of another.