What is the spin rotation operator for spin > 1/2? For spin $\frac{1}{2}$, the spin rotation operator $R_\alpha(\textbf{n})=\exp(-i\frac{\alpha}{2}\vec{\sigma}\cdot\textbf{n})$ has a simple form: 
$$R_\alpha(\textbf{n})=\cos\biggl(\frac{\alpha}{2}\biggr)-i\vec{\sigma}\cdot\textbf{n}\sin\biggl(\frac{\alpha}{2}\biggr)$$
What about spin > $\frac{1}{2}$ ?
 A: What one has for the spin 1/2, 
\begin{equation}
 \exp(i\alpha \mathbf{J}\cdot\hat{\mathbf{n}}) = \cos(\alpha/2) + i\sin(\alpha/2)\mathbf{J}\cdot\hat{\mathbf{n}},
\end{equation}
as well for spin 1 through Rodrigues formula,
\begin{equation}
 \begin{aligned}
  \exp(i\alpha \mathbf{J}\cdot\hat{\mathbf{n}}) & = 1 + i\hat{\mathbf{n}}\cdot\mathbf{J}\sin\alpha + (\hat{\mathbf{n}}\cdot\mathbf{J})^2(\cos\alpha-1) \\
& = 1 + \left[2i\hat{\mathbf{n}}\cdot\mathbf{J}\sin(\alpha/2)\right]\cos(\alpha/2) + \frac{1}{2}\left[2i\hat{\mathbf{n}}\cdot\mathbf{J}\sin(\alpha/2)\right]^2,
 \end{aligned}
\end{equation}
is a spin representation of rotation operators as finite order polynomials of the rotation generators for $j=1/2,1$, where the coefficients are sines and cosines of half the angle of rotation. It was known that this could be extended for higher spin representations, but the exact polynomial expression for any spin $j$ remained unknown. Fortunately, in 2014 this general expression was found by Curtright, Fairlie & Zachos. I leave here their publication: http://arxiv.org/abs/1402.3541
A: The same, except that the $\sigma_k$ are now not Pauli matrices but the generators of a su(2) representation of the desired spin. For example, the $3\times 3$ matrices 
$$ \sigma_\ell:=(2\epsilon_{jk\ell})_{j,k=1:3}$$
define the spin 1 representation on 3-vectors. [Maybe the factor 2 should take a different value.] The corresponding explicit formula comes from the Rodrigues formula
$$e^{X(a)}=1+\frac{\sin|a|}{|a|}X(a)+\frac{1-\cos|a|}{|a|}X(a)^2,$$
where $X(a)$ is the matrix mapping a vector $b$ to $X(a)b=a \times b$.
For higher spin, the corresponding formula will depend on how you write the representation. Numerically, one would just diagonalize the matrix in the exponent; then computing the exponential is trivial. I don't know whether for general spin if there is any advantage in having an explicit formula.
