  Let $G$ be a finite group and $R$ be a commutative ring. The Mackey algebra
$\mu_{R}(G)$ shares a lot of properties with the group algebra $RG$ however,
there are some differences. For example, the group algebra is a symmetric
algebra and this is not always the case for the Mackey algebra. In this paper
we present a systematic approach to the question of the symmetricity of the
Mackey algebra, by producing symmetric associative bilinear forms for the
Mackey algebra. The category of Mackey functors is a closed symmetric monoidal
category, so using the formalism of J.P. May for these categories, S. Bouc has
defined the so-called Burnside trace. Using this Burnside trace we produce
trace maps for Mackey algebras which generalize the usual trace map the group
algebras. These trace maps factorise through Burnside algebras. We prove that
the Mackey algebra $\mu_{R}(G)$ is a symmetric algebra if and only if the
family of Burnside algebras $(RB(H))_{H\leqslant G}$ is a family of symmetric
algebras with a compatibility condition. As a corollary, we recover the well
known fact that over a field of characteristic zero, the Mackey algebra is
always symmetric. Over the ring of integers the Mackey algebra of $G$ is
symmetric if and only if the order of $G$ is square free. Finally, over a field
of characteristic $p>0$ we show that the Mackey algebra is symmetric if and
only if the Sylow $p$-subgroups of $G$ are of order $1$ or $p$.
