There are two old books, authored by N.P. Sokolov, and available in Russian only:
"Spatial matrices and their applications", Gos. Izdat. Fiz.-Mat. Lit., Moscow, 1960 [MR: 0130256] and "Introduction to the theory of multidimensional matrices", Naukova Dumka, Kiev, 1972 [MR: 0352115]. At least the first of these books is available on the web.
From the MR review of the first book:
Such arrays were first considered by Cayley [Trans. Cambridge Philos. Soc. 8 (1842/49), 75–88; pp. 85–88] and have since been the subject of numerous investigations. Of the considerable literature that has grown up in this field, we may mention a long series of papers by M. Lecat published between 1910 and 1929 and also, in more recent years, the work of R. Oldenburger.
I guess this topic is out of fashion now.