Orthogonality of matrices in the Ky Fan $k$-norms

We obtain necessary and sufficient conditions for a matrix $A$ to be Birkhoff-James orthogonal to another matrix $B$ in the Ky Fan $k$-norms. A characterization for $A$ to be Birkhoff-James orthogonal to any subspace $\mathcal{W}$ of $\mathbb{M}(n)$ is also obtained.