  The half-maximal supergravity theories in three dimensions, which have local
$SO(8)\xz SO(n)$ and rigid SO(8,n) symmetries, are discussed in a superspace
setting starting from the superconformal theory. The on-shell theory is
obtained by imposing further constraints; it is essentially a non-linear sigma
model that induces a Poincar\'e supergeometry. The deformations of the geometry
due to gauging are briefly discussed. The possible $p$-form field strengths are
studied using supersymmetry and SO(8,n) symmetry. The set of such forms obeying
consistent Bianchi identities constitutes a Lie super co-algebra while the
demand that these identities admit solutions places a further constraint on the
possible representations of SO(8,n) that the forms transform under which can be
easily understood using superspace cohomology. The dual Lie superalgebra can
then be identified as the positive sector of a Borcherds superalgebra that
extends the Lie algebra of the duality group. In addition to the known
$p=2,3,4$ forms, which we construct explicitly, there are five-forms that can
be non-zero in supergravity, while all forms with $p>5$ vanish. It is shown
that some six-forms can have non-trivial contributions at order $\a'$.
