Certain part in my textbook implies that a representation of $SO(3)$ in the vector space $V = \mathbb C^{2S+1}$, where $S \in \mathbb Z$, is possible. I am trying to find a path that leads to this representation. My reasoning is the following.
The induced Lie algebras $so(3)$ and $su(2)$ are isomorphic, so I can uniquely map an element of $so(3)$ into $su(2)$. Now by considering the representation of $SU(2)$ in a space of polynomial of fixed degree $l$ on $\mathbb C^2$, denoted by $P_l(\mathbb C^2)$, I can show that the induced Lie algebra $su(2)$ in $P_l(\mathbb C^2)$ has matrix form of size $(2S+1)\times(2S+1)$. Thus, it can be a candidate for the representation of $so(3)$ in $V = \mathbb C^{2S+1}$, generating the representation of $SO(3)$ in this space.
Is my answer correct?
Your answer certainly goes in the right direction, but you are missing one point. You should use the action of $SU(2)$ on homogeneous polynomials of degree $\ell$. Initially, this can be done for any degree $\ell$ and the resulting space has dimension $\ell+1$. (A basis is formed by the elements $z^iw^{\ell-i}$ for $i=0,\dots,\ell$.) This gives you a representation of $\mathfrak{su}(2)\cong\mathfrak{so}(3)$ in any complex dimension. Now it remains to show that in odd dimensions, i.e. in the case of homogeneous polynomials of even degree, there is a corresponding representation of the group $SO(3)$. To do this, you have to observe that the isomorphism of the Lie algebras corresponds to the fact that there is a surjective homomorphism $SU(2)\to SO(3)$ with kernel $\{\pm\mathbb I\}$. This implies that representations of $SO(3)$ correspond to representations of $SU(2)$ on which $-\mathbb I$ acts trivially. In the case of homogeneous polynomials of degree $\ell$, $-\mathbb I$ acts by multiplication by $(-1)^\ell$, which implies the claim.