I have seen written that the space of $m\times m$-complex matrices $M_m(\mathbb C)$ endowed with the usual Jordan product is isomorphic, as a Jordan algebra to the complexification of the space $Herm(m,\mathbb C)$ of hermitian $m\times m$ matrices.

Is it possible (easy) to write down an explicit isomorphism between the two?

For example, for the $2\times 2$ matrices, the elements of $Herm(2,\mathbb C)\otimes_{\mathbb R}\mathbb C$ are of the form $\left(\begin{smallmatrix}\alpha &\lambda + I\mu\\\lambda-I\mu &\beta\end{smallmatrix}\right)$ with $I=i\otimes_{\mathbb R}1$ and $\alpha,\beta,\lambda,\mu\in \mathbb C$ and the elements of $M_2(\mathbb C)$ are of the form $\left(\begin{smallmatrix}\alpha &c\\d &b\end{smallmatrix}\right)$ with $a,b,c,d\in \mathbb C$. The question is: is there a nice expression of $a,b,c,d$ in terms of $\alpha,\beta,\lambda,\mu$?