The determinant as a volume

Triple product. The scalar triple product $u\cdot(v\times w)$ equals $\det(u,v,w)$ (with $u,v,w$ as columns). Indeed, the cross product $v\times w$ has norm equal to the area of the parallelogram spanned by $v,w$, and is orthogonal to the plane of $v,w$. So $u\cdot(v\times w)$ is the volume of the parallelepiped spanned by $u,v,w$.

Determinant. The scalar triple product can be written as a determinant:
$$u\cdot(v\times w) = \begin{vmatrix} u_1 & v_1 & w_1 \\ u_2 & v_2 & w_2 \\ u_3 & v_3 & w_3 \end{vmatrix} = \det(u,v,w).$$

Conclusion. $|\det(u,v,w)|$ is the volume of the parallelepiped.

Higher dimension. In $\mathbb{R}^n$, the same holds: $|\det(u_1,\ldots,u_n)|$ is the $n$-dimensional volume of the parallelotope spanned by the vectors.