$SU(2)$ representation that conserves the Lie algebra

Question

I've seen that there are many representations of $SU(2)$ on $\mathbb R^3$, one of these representation is the map from $SU(2)$ to the set of matrices of $SO(3)$. This representation conserves the Lie algebra of SU(2) (I mean that the imagine of the map is a group with the same Lie algebra of the domain). My question is, are there other representations of $SU(2)$ on $\mathbb R^3$ which image is a group with the same Lie algebra of $SU(2)$?