The determinant of a matrix can be interpreted as the signed volume of the unit square/cube/hypercube, etc.

Taking the latter as the definition of the determinant, I can understand why, in the $2\times2$ case, the determinant is computed in such a way, yet why is it that when dealing with a $3\times3$ matrix (or more generally, a $n \times n$ matrix) the determinant is computed the way it is?

Would appreciate any help!