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
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
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.