How to get results in terms of vector operations?

Question

Is there a simple way to simplify an expression in terms of vector operations?

For example, when I evaluate this;

v1 = {x1, y1, z1};
v2 = {x2, y2, z2};
v3 = {x3, y3, z3};
v4 = {x4, y4, z4};
Integrate[1, {x, y, z} \[Element] Tetrahedron[{v1, v2, v3, v4}]]

I get this horrible abomination;

1/6 Abs[x3 y2 z1 - x4 y2 z1 - x2 y3 z1 + x4 y3 z1 + x2 y4 z1 - x3 y4 z1 - x3 y1 z2 + x4 y1 z2 + x1 y3 z2 - x4 y3 z2 - x1 y4 z2 + x3 y4 z2 + x2 y1 z3 - x4 y1 z3 - x1 y2 z3 + x4 y2 z3 + x1 y4 z3 - x2 y4 z3 - x2 y1 z4 + x3 y1 z4 + x1 y2 z4 - x3 y2 z4 - x1 y3 z4 + x2 y3 z4]

However, this is simply the formula for the tetrahedron volume;

$$\frac16 \left| ( \vec{v}_2 - \vec{v}_1 ) \cdot ( ( \vec{v}_3 - \vec{v}_1 ) \times ( \vec{v}_4 - \vec{v}_1 ) ) \right|$$

Can Mathematica show the result I get in terms of vectors and vector operations?

There are other questions similar to this, but answers are some hacky manipualtions and not quite what I'm looking for.

Isn't there a simple, non-hacky, built-in way? There must be! C'mon Mathematica...