We study in this paper properties of Schur multipliers of Schatten von Neumann classes S_p. We prove that for p≤1, Schur multipliers of S_p are necessarily completely bounded. We also introduce for p≤1 a scale W_p of tensor products of ℓ^∞ and prove that matrices in W_p are Schur multipliers of S_p. We compare this sufficient condition with the sufficient condition of membership in the p-tensor product of ℓ^∞ spaces.