  Let $V$ be a finite-dimensional vector space over a field of characteristic
two. As the main result of this paper, for every nilpotent element $e \in
\mathfrak{sl}(V)$, we describe the Jordan normal form of $e$ on the
$\mathfrak{sl}(V)$-modules $\wedge^2(V)$ and $S^2(V)$. In the case where $e$ is
a regular nilpotent element, we are able to give a closed formula. We also
consider the closely related problem of describing, for every unipotent element
$u \in \operatorname{SL}(V)$, the Jordan normal form of $u$ on $\wedge^2(V)$
and $S^2(V)$. A recursive formula for the Jordan block sizes of $u$ on
$\wedge^2(V)$ was given by Gow and Laffey (J. Group Theory 9 (2006), 659-672).
We show that their proof can be adapted to give a similar formula for the
Jordan block sizes of $u$ on $S^2(V)$.
