Abstract: Every element of M_n(ℤ_2p+1)​ can be written as a sum of orthogonal matrices in M_n(ℤ_2p+1)​.  We show that not every element of M₂(ℤ₄)​ can be written as a sum of orthogonal matrices.  We also show that a matrix A​ in M₃(ℤ₂)​ can be written as a sum of orthogonal matrices if and only if each column sum and each row sum of A​ are equal.