For a $n\times n$ matrix, the volume is given by its determinant. In the case where $n=2$ the determinant is the area, in the case where $n=3$ it is the volume, and it generalizes to $n$ dimensions.
My question; Is there an equivalent to a volume formula/concept for a tensor? I have in mind a $4\times 4 \times 4 \times 4$ tensor if it helps.