  An Orthogonally resolvable Matching Design OMD$(n, k)$ is a partition of the
edges the complete graph $K_n$ into matchings of size $k$, called blocks, such
that the blocks can be resolved in two different ways. Such a design can be
represented as a square array whose cells are either empty or contain a
matching of size $k$, where every vertex appears exactly once in each row and
column. In this paper we show that an OMD$(n.k)$ exists if and only if $n
\equiv 0 \pmod{2k}$ except when $k=1$ and $n = 4$ or $6$.
