Intuitively, does the wedge product describe the oriented area?

Question

Is the following interpretation correct:

Given two vectors in $\mathbb{R}^n$, their wedge product describes an oriented area with magnitude equal to the area of the parallelogram formed by the two vectors and orientation perpendicular to both?

Answer 1

Almost correct, except for “orientation perpendicular to both”.

If $n>3$, there are infinitely many directions perpendicular to the two vectors, so you can't think of the orientation as a vector (like you do for the cross product in three dimensions).

Instead, you may think of the orientation as a circle in the plane of the two given vectors $\mathbf{u}$ and $\mathbf{v}$, with a direction attached to it in one of the two possible ways: $\circlearrowleft$ or $\circlearrowright$.

The orientation corresponding to $\mathbf{u} \wedge \mathbf{v}$ is the direction in which you should rotate the first factor $\mathbf{u}$ to align it with the second factor $\mathbf{v}$. (The shortest possible rotation, of course, not the detour the other way around the circle.)

To align $\mathbf{v}$ with $\mathbf{u}$ instead, you must rotate in the opposite direction, so you get the opposite orientation, which algebraically corresponds to a change of sign: $\mathbf{v} \wedge \mathbf{u} = -\mathbf{u} \wedge \mathbf{v}$.

(For wedge products with more than two factors, it's not quite as simple. For $\mathbf{u} \wedge \mathbf{v} \wedge \mathbf{w}$, the two possible orientations correspond to whether $(\mathbf{u},\mathbf{v},\mathbf{w})$ form a right-handed or a left-handed system in the three-dimensional subspace of $\mathbb{R}^n$ that they span.)