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}