  We give a general construction leading to different non-isomorphic families
$\Gamma_{n,q}(\K)$ of connected $q$-regular semisymmetric graphs of order
$2q^{n+1}$ embedded in $\PG(n+1,q)$, for a prime power $q=p^h$, using the
linear representation of a particular point set $\K$ of size $q$ contained in a
hyperplane of $\PG(n+1,q)$. We show that, when $\K$ is a normal rational curve
with one point removed, the graphs $\Gamma_{n,q}(\K)$ are isomorphic to the
graphs constructed for $q$ prime in [9] and to the graphs constructed for
$q=p^h$ in [20]. These graphs were known to be semisymmetric but their full
automorphism group was up to now unknown. For $q\geq n+3$ or $q=p=n+2$, $n\geq
2$, we obtain their full automorphism group from our construction by showing
that, for an arc $\K$, every automorphism of $\Gamma_{n,q}(\K)$ is induced by a
collineation of the ambient space $\PG(n+1,q)$. We also give some other
examples of semisymmetric graphs $\Gamma_{n,q}(\K)$ for which not every
automorphism is induced by a collineation of their ambient space.
