Let $A \in \mathbb{R}^{N \times N}$ be a D6160: Real square matrix.
Let $S_N$ be a D4951: Set of standard N-permutations.
The determinant of $A$ is the D993: Real number $$\begin{equation} \text{Det} A : = \sum_{\pi \in S_N} \left( \text{Sign}(\pi) \prod_{n = 1}^N A_{n, \pi(n)} \right) \end{equation}$$

Let $A \in \mathbb{R}^{N \times N}$ be a D6160: Real square matrix.
Let $S_N$ be a D4951: Set of standard N-permutations.
The determinant of $A$ is the D993: Real number $$\begin{equation} \text{Det} A : = \sum_{\pi \in S_N} \text{Sign}(\pi) A_{1, \pi(1)} A_{2, \pi(2)} A_{3, \pi(3)} \cdots A_{N - 1, \pi(N - 1)} A_{N, \pi(N)} \end{equation}$$