Show that R is a rotation of the Bloch sphere

Question

I've shown that $$R=e^{-i\frac{\theta}{2}X} = \cos\left(\frac{\theta}{2}\right)I - i\sin\left(\frac{\theta}{2}\right)X$$ Where $X$ is the matrix $\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$, and this would apply to any other matrix $X$ such that $X^2=I$ like the other Pauli matrices. But from this it's not clear to me that $R$ is a rotation or what the axis or angle of rotation is, and I wouldn't know where to start with the calculation suggested by the first answer in this question. Is there an alternative approach I could take?