Four dimensional cross product of THREE vectors There are many MSE posts about how to define a cross product in $\mathbb{R^4}$. It is impossible to define a cross product of two vectors in $\mathbb{R^4}$, since there are infinitely many directions perpendicular to those two vectors, and we don't know which direction to choose. However, If we are given THREE vectors $A,B,C$, it is possible to find a unique direction perpendicular to this three vectors, if $A,B,C$ are independent. However, finding this perpendicular vector involves solving a system of equations.
So my question is：can we define a Quasi Cross Product $\{A,B,C\}$ on $\mathbb{R^4}$, so that we can find a direction perpendicular to $A,B,C$ without solving a system of equations?

The short answer is yes. One way is to take the formal determinant
$$\left|\begin{matrix}e_1&e_2&e_3&e_4\\
a_1&a_2&a_3&a_4\\
b_1&b_2&b_3&b_4\\
c_1&c_2&c_3&c_4\\
\end{matrix}\right|$$
where $e_1,\ldots,e_4$ are the standard unit vectors, and $a=\sum a_ie_i$
etc., are the three vectors.
Or you can rephrase this in terms
of exterior powers and the Hodge star operator.
All this works in $n$ dimensions too.
