Let $S$ be a real closed Riemann surfaces together a reflection
\mbox{$\tau:S \to S$}, that is, an anticonformal involution with
fixed points. A well known fact due to C. L. May \cite{May 1977}
asserts that the group $K(S,\tau)$, consisting on all
automorphisms (conformal and anticonformal) of $S$ which commutes
with $\tau$, has order at most $24(g-1)$. The surface $S$ is
called maximally symmetric Riemann surface if
$|K(S,\tau)|=24(g-1)$ \cite{Greenleaf-May 1982}. In this note we
proceed to construct real Schottky uniformizations of all
maximally symmetric Riemann surfaces of genus $g \leq 5$. A method
due to Burnside \cite{Burnside 1892} permits us the computation of a
basis of holomorphic one forms in terms of these real Schottky
groups and, in particular, to compute a Riemann period matrix for
them. We also use this in genus 2 and 3 to compute an
algebraic curve representing the uniformized surface $S$. The
arguments used in this note can be programed into a computer
program in order to obtain numerical approximation of Riemann
period matrices and algebraic curves for the uniformized surface
$S$ in terms of the parameters defining the real Schottky groups.