I've seen many sources about the geometric meaning of determinants, such as the 3blue1brown's "Essence of Linear Algebra" playlist and this video. However, I couldn't find anywhere the relationship between the algebraic expression for calculating the value of the determinant of an $ n \times n$ matrix and the signed "hypervolume" of the unit cube of $ \mathbb{R^n} $ after the transformation done by the $n \times n$ matrix.

At most, the sources would only cover cases such as $n=1,2 \text{ or } 3$, which are easy to be dealt with. But what I really would like to know is how the full algebraic expression for calculating the determinant of an $n \times n$ matrix connects to the signed volumes of transformed hypercubes of $\mathbb{R}^n$ for all $n \in \mathbb{Z}^+_*$.

**Clarification:** by *algebraic expression for calculating the determinant* I mean the use of something like Laplace's expansion $\det(A_{n \times n}) = \sum\limits_{i=1}^n (-1)^{i+j} M_{ij}$. So what I want is to, given an intuitive definition of something like a "hypervolume" of the "parallelogram" at $\mathbb{R}^n$ whose points include the origin and each of the matrice's columns, arrive at this expression for the determinant.