For determinants of $2 \times 2$ and $3 \times 3$ matrices, there is a simple geometric interpretation of the determinant. It is, respectively, the area of parallelogram or parallelepiped formed by the transformed basis vectors of a vector space following a linear transformation.
For determinants of $n \times n$ matrices with $n \geq 4$, there is obviously no simple geometric interpretation. One possible interpretation for the determinants of these higher order matrices involves linear systems of equations, where, for a system of $n$ linear equations, if the coefficient matrix $A$ has a determinant det$(A) = 0$, the system has no solution.
Are there any other interesting interpretations or applications of higher order determinants?