  We introduce the concept of n-OU and n-OO matrix sets, a collection of n
mutually-orthogonal unitary and real orthogonal matrices under Hilbert-Schmidt
inner product. We give a detailed characterization of order-three n-OO matrix
sets under orthogonal equivalence. As an application in quantum information
theory, we show that the minimum and maximum numbers of an unextendible
maximally entangled bases within a real two-qutrit system are three and four,
respectively. Further, we propose a new matrix decomposition approach, defining
an n-OU (resp. n-OO) decomposition for a matrix as a linear combination of n
matrices from an n-OU (resp. n-OO) matrix set. We show that any order-d matrix
has a d-OU decomposition. As a contrast, we provide criteria for an order-three
real matrix to possess an n-OO decomposition.
