Let $G:=\mathrm{Sp}_{2n}$ be the simple algebraic group of simply connected type with root-system $C_n$, over a field $K$ of characteristc $2$. Is there a way, to explicitly construct the matrices corresponding to the highest weight representation $\mathrm{L}(\lambda)$, where $\lambda \in X(T)$ is the fundamental dominant weight corresponding to the shortest root in the Basis?

This should be the highest weight module of dimension $\dim{\mathrm{L}(\lambda)}=2^n$, where $n$ is the rank of the group.

So I'm asking for the representation of $G(\mathbb{F}_2)$.

I need these matrices for $n\geq 7$ for some computations.