While I'm not sure this question is appropriate for this site, here goes.
First, you need a maximal torus. Inside SO(5) we have SO(4) and then SO(2) x SO(2).
Write your antisymmetric matrices as
$$\begin{matrix} aJ & C & v \end{matrix} $$ $$\begin{matrix} -C^T & bJ & w \end{matrix} $$ $$\begin{matrix} -v^T & -w^T & 0 \end{matrix} $$ where $J = \left( {0\atop -1}{1\atop 0} \right) $, $C$ is square, and $v$ and $w$ are columns. Then the $a$ and $b$ parts are the torus, the $v$ gets you the $\pm x_1$ weights, the $w$ gets you the $\pm x_2$, and the $C$ gets you the $\pm x_1\pm x_2$.
Taking $x_1$ and $x_2 - x_1$ as simple roots, the $e_{x_1}$ is \begin{pmatrix} 0&0&0&0&1\\ 0&0&0&0&i\\ 0&0&0&0&0\\ 0&0&0&0&0\\ -1&-i&0&0&0 \end{pmatrix} and the $e_{x_2-x_1}$ is \begin{pmatrix} 0&0&1&i&0\\ 0&0&i&-1&0\\ -1&-i&0&0&0\\ -i&1&0&0&0\\ 0&0&0&0&0 \end{pmatrix}
