The basic theory here (for $p=2$) was worked out by R. Steinberg in his 1963 *Nagoya Math. J.* paper on representations of algebraic groups. On the other hand, computing explicit matrices for rank at least 7 promises to be cumbersome, unless you limit attention to key operators in the group corresponding to roots, etc. (Frank Lubeck and his collaborators have probably done the most sophisticated computing, but usually with a focus on basic root and weight data.)

Your formulation needs a little more care. It's true here that the simply connected algebraic groups of types $B_n, C_n$ are related in characteristic 2 by one of Chevalley's special isogenies; this in turn makes the underlying abstract groups over an algebraically closed field, or a finite field, isomorphic. Steinberg's general study of irreducible representations shows that these are realized as "twisted" tensor products of a basic collection belonging to the finite group over the prime field (or a corresponding restricted Lie algebra). So it's possible to study the representations of interest to you by working only with the finite groups over $\mathbb{F}_2$.

Steinberg's ideas are explained (carefully, I hope) in Sections 5.3-5.4 of my LMS Lecture Note volume *Modular Representations of Finite Groups of Lie Type* (2006). In particular, the theory leads to collections of modules supported on only long or only short roots in a precise way. This is illustrated at the end of my 5.4 for the groups $B_3, C_3$ and their three fundamental weights. The dimensions of irreducibles coincide and are respectively $6, 14, 8$. In general you do get $2^n$ for the last one, coming essentially from type $B_n$ (where there is a unique short simple root numbered $\alpha_n$). .

Short of doing computer calculations for such groups, there is some useful literature summarized in Section 4.5 of my book. This is due to Premet-Suprunenko, McNinch, and Foulle in particular.

P.S. I should add that the fundamental weight for $B_n$ which gives rise to the dimension $2^n$ is *minuscule*, so there is a single orbit of weights under the Weyl group. In particular, a basis for the representation space is uniquely determined up to scalars, though it's still so big that explicit matrices become almost impossible to write down as $n$ increases.