Spin structure on Whitney sum

Question

Suppose $E_1$ and $E_2$ are oriented vector bundles with $w_2(E_i)=0$ over a compact manifold $M$. Then $E=E_1 \oplus E_2$ admits a spin structure too. Can we choose a spin structure of $E$ such that the associated spin vector bundle $S(E)$ is given as the Whitney sum of $S(E_1)\oplus S(E_2)$?