Consider the symplectic Lie algebra $\mathfrak{s} \mathfrak{p}_n$ over a field $K$.
I know that the root system is given by $C_n=\{\pm 2e_j, \pm e_j \pm e_k:j,k=1 \cdots n, j \neq k\} $
where $e_k(diag(h,-h))=h_k$ with $h \in K$.
I do not know what are the corresponding root spaces for each root. Thank you for discussing the root spaces (possibly with the help of matrices, diagrams so that it is easier for me to understand).
Denote the positive roots $a_{ij} = e_i - e_j$ (for $i<j$), $b_i = 2e_i$, $c_{ij} = e_i + e_j$ (for $i<j$). Then the root subspaces correspond to the matrix enries in the following way:
$\begin{equation} \left(\begin{array}{lllll|lllll} \ast & a_{12} & a_{13} & \cdots & a_{1n} & b_1 & c_{12} & c_{13} & \cdots & c_{1n} \\ -a_{12} & \ast & a_{23} & \cdots & a_{2n} & c_{12} & b_2 & c_{23} & \cdots & c_{2n} \\ -a_{13} & -a_{23} & \ast & \cdots & a_{3n} & c_{13} & c_{23} & b_3 & \cdots & c_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ -a_{1n} & -a_{2n} & -a_{3n} & \cdots & \ast & c_{1n} & c_{2n} & c_{3n} & \cdots & b_n \\ \hline % -b_1 & -c_{12} & -c_{13} & \cdots & -c_{1n} & \ast & -a_{12} & -a_{13} & \cdots & -a_{1n} \\ -c_{12} & -b_2 & -c_{23} & \cdots & -c_{2n} & a_{12} & \ast & -a_{23} & \cdots & -a_{2n} \\ -c_{13} & -c_{23} & -b_3 & \cdots & -c_{3n} & a_{13} & a_{23} & \ast & \cdots & -a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ -c_{1n} & -c_{2n} & -c_{3n} & \cdots & -b_n & a_{1n} & a_{2n} & a_{3n} & \cdots & \ast \end{array} \right) \end{equation}.$
Recall that a matrix from $\mathfrak{sp}$ must have its upper-right and lower-left block symmetric, and that lower-right block is negative transpose of upper left. So root subspaces are one dimensional.