When hermitian operators $L_1, L_2, L_3$ follow the commutation relations:
$$ [L_1,L_2]=i\;L_3 \\ [L_2,L_3]=i\;L_1 \\ [L_3,L_1]=i\;L_2 $$
one can show that, assuming they are in finite number, their eigenvalues are integer or half-integer. It turns out that the basis of the Lie algebra of $SU(2)$ follows these relations, and after identifying the angular momentum to these operators, it constitutes a proof of the quantization of angular momentum.
Now, my mathematical knowledge does not go much further, but the basis of the Lie algebra of $SO(3)$ follows the same commutation relations, with the exception of the $i$ coefficient. In that case, I cannot find any way to obtain quantized eigenvalues. How come I read here and there that $SO(3)$ and $SU(2)$ have isomorphic Lie algebras, and that they basically lead to the same quantization? If so, how do I derive that quantization using only $SO(3)$ operators? And if the result is the same, why would we bother using $SU(2)$ when we can use real 3x3 rotation operators?