Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix.
Let $S_N$ be a D4951: Set of standard N-permutations.
The determinant of $A$ is the D1207: Complex number
\begin{equation}
\text{Det} A
: = \sum_{\pi \in S_N} \text{Sign}(\pi) A_{1, \pi(1)} A_{2, \pi(2)} A_{3, \pi(3)} \cdots A_{N - 1, \pi(N - 1)} A_{N, \pi(N)}
\end{equation}