Parameterization of rotations in three-dimensional space

Question

I am dealing with 3 dimentional representation of $\mathfrak{su}(2)=\langle iJ_x, iJ_y, iJ_z\rangle$: $$ J_x=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \quad J_y=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{pmatrix}, \quad J_z=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}.$$ I want to parameterize arbitrary rotation in 3D using Euler angles and this representation. Is it right to do it as follows $$\mathcal{R} = \mathcal{R}(\alpha, \beta, \gamma) = \exp{[-i\alpha J_z]}\cdot \exp{[-i\beta J_x]}\cdot \exp{[-i\gamma J_z]}?$$