Is there a name for "multiplicatively skew-symmetric matrices" and what are good techniques for computing their determinant?

Question

In the course of a counting problem related to graph paths, I encounter a type of matrix that satisfies the following properties:

1. All diagonal elements are zero, that is, $a_{ii} = 0$ for all i
2. Elements across the diagonal are inverse of each other, that is, $a_{ij} = a_{ji}^{-1}$ for all $i,j$

The definition looks related to skew symmetric matrices. Do these matrices have a name? Do you know of specific methods for computing their determinant?

I have looked at a list of matrix classes on Wikipedia in surveys of matrix classes without success. If that helps, in my case, the matrices only have 3 possible values for non-diagonal entries: $1$, $x$ and $x^{-1}$.