A particular representation of $SU(2)$ on $\mathbb R^3$

Question

Studying physics I encounter group theory and it has told me that: the matrices that rotates $\mathbb R^3$ vectors in the Euclidean space are the representation of $SU(2)$. Namely, $SO(3)$ matrices are the representation of $SU(2)$ on $\mathbb R^3$.

After they told me this I searched for the definition on representation and I thought that the map to the identity should also be a representation of $SU(2)$ on $\mathbb R^3$, namely:

if $g \in SU(2)$ and $f$ is a map such that $f(g)=\begin {bmatrix} 1&0&0\\0&1&0\\0&0&1\end {bmatrix}$ then $f$ is a representation of $SU(2)$ on $\mathbb R^3$.

If this is true it means that there are many representation of $SU(2)$ on $\mathbb R^3$ and one of these gives us the matrices of $SO(3)$. Then what is special in the representation that gives us the matrices of $SO(3)$?

I've tried to answer this question by myself but the only thing I came up with is that the representation that gives the $SO(3)$ group has the same Lie algebra of $SU(2)$. However I'm not able to check if the representation of $SU(2)$ on $\mathbb R^3$ that gives the $SO(3)$ group is the only one with this Lie algebra.

Answer 1

OP explicitly considers the trivial representation $SU(2)\to GL(3,\mathbb{R})$. However, modulo equivalence of representations, the sought-for irreducible representation $\rho:SU(2)\to GL(3,\mathbb{R})$ is undoubtedly $\rho=R\circ\pi$, where $\pi:SU(2)\to SO(3)\cong SU(2)/ \mathbb{Z}_2$ is the covering homomorphism, and $R:SO(3)\to GL(3,\mathbb{R})$ is the defining representation for $SO(3)$, aka. the vector representation.