I have a matrix that is “kind of symmetric.” Specifically, it is an $n \times n$ real matrix such that the entries $a_{ij}=1/a_{ji}$ whenever $j \ne i$. I want to investigate the properties of this matrix. As a starting place, does this kind of matrix ( or perhaps the matrix where $a_{ii}=1$ as well) have a special name and has it been studied?

I have a matrix that is “kind of symmetric.” Specifically, it is an $n \times n$ real matrix such that the entries $a_{ij}=1/a_{ji}$ whenever $j \ne i$. I want to investigate the properties of this matrix. As a starting place, does this kind of matrix (or perhaps the matrix where $a_{ii}=1$ as well) have a special name and has it been studied?