Problem #17 in Zhan's survey of open problems in matrix theory is the Li-Poon problem on writing a square real matrix as the linear combination of $k$ orthogonal matrices. They proved that it is possible to take $k=4$ for every square real matrix and asked if $k$ is the least possible such number.

I am wondering if there is a ring-theoretical version of this conjecture. This raises some even more basic questions:

What is the ring-theoretic equivalent of an orthogonal matrix?

And since an orthogonal matrix $O$ can be characterized as satisfying $OO^{T}=I$,

What is the ring-theoretic equivalent of the matrix transpose operation?