can we expand the relation between the determinant of a transformation and area of parallelogram to farther dimensions?

Question

I'm now trying to understand determinants of linear transformations, and I was introduced to determinants as (the scaling factor that we multiply it by any area to figure out what the area of the image of the first area is going to be under a transformation), and I was told that the same thing goes when dealing with 3-dimensional transformations (but this time with volumes). So my question is can we extend that relation between determinants and shapes to further (4,5,6,...,etc.) dimensions? and, if we can, then how can we relate the formula of $n \times n$ matrices’ determinants $$\det(A)=\sum_{j=1}^n (-1)^{i+j} a_{ij}\det(A_{ij})$$(where $j$ is the column (unfixed) and $i$ is the row (fixed) and $A_{ij}$ is the $n-1 \times n-1$ matrix that remains after removing the $i^{th}$ row and the $j^{th}$ column)?