Let $a_{1}, a_{2}, a_{3} \in \mathbb{R}^{3}(\mathbb{R})$ form a linearly independent set. It is well-known that the parallelepiped $P$ having $a_{1}, a_{2}, a_{3}$ as adjacent sides has volume equal to the determinant of the matrix whose rows are $a_{1}, a_{2}, a_{3}$.
But I wonder if there is any labor-saving way to calculate $P$ without using determinant in general? I say "in general" because if $a_{1}, a_{2}, a_{3}$ happen to be vertical to each other then the volume of $P$ equals simply $|| a_{1} ||\cdot ||a_{2}||\cdot ||a_{3}||.$