The determinant is basically the cross product – see the interaction between dimensions. This tracks area by seeing how much each dimension is pushing on the other.
The orientation is signed, so once way (low -> high is positive, high -> low is negative). That’s why the cross product can be seen as the determinant of the 3-element matrix.
Nice discussion here: https://www.reddit.com/r/math/comments/3mag55/can_we_have_a_post_that_is_teach_me_something/cvdgd31