Can any bipartite state be written as $\rho_{AB}=\sum_{ij} p_{ij}\rho_A^i\otimes \rho_B^j$?

I'm having a silly doubt. If $\rho_{AB}$ is separable then we can write it as $\rho_{AB}=\sum_i p_i\rho_A^i\otimes \rho_B^i$ but can we write a general (maybe entangled) state as $\rho_{AB}=\sum_{ij} p_{ij}\rho_A^i\otimes \rho_B^j$ ?