determinant as signed volume

The cleanest mental picture of a determinant: feed it the n columns of a matrix, and it returns the signed n-dimensional volume of the box (parallelepiped) those columns span. In 2D it is the signed area of the parallelogram on two vectors; in 3D the signed volume of the parallelepiped on three. 'Signed' means it also records orientation by a plus or minus.

Because det is the signed volume of the image of the unit cube, det(A) is precisely the factor by which the linear map A scales every volume: a region of volume V is sent to a region of volume |det(A)| * V. If det(A) = 0 the cube is squashed flat into a lower dimension (the columns are dependent); if det(A) is negative, the map turns the box inside out, reversing handedness.

This geometric reading explains the algebra effortlessly. Multilinearity is 'doubling one edge doubles the volume.' The alternating property is 'two equal edges means a flat, zero-volume box.' Multiplicativity det(AB) = det(A) det(B) is 'do one volume-scaling, then another, and the scalings multiply.' You can re-derive the rules by drawing pictures.

Caveat: 'volume' here is the natural (Lebesgue) volume tied to the standard basis; the sign depends on a chosen orientation. The number det(A) is basis-free for an operator, but the very notion of 'unit volume 1' requires you to declare a reference box. Change that reference and the absolute scale of 'volume' changes, though ratios of volumes — and hence det of an operator — do not.