  Let $G$ be a finite group and $k$ an algebraically closed field of
characteristic $p>0$. We define the notion of a Dade $kG$-module as a
generalization of endo-permutation modules for $p$-groups. We show that under a
suitable equivalence relation, the set of equivalence classes of Dade
$kG$-modules forms a group under tensor product, and the group obtained this
way is isomorphic to the Dade group $D(G)$ defined by Lassueur. We also
consider the subgroup $D^{\Omega} (G)$ of $D(G)$ generated by relative syzygies
$\Omega_X$, where $X$ is a finite $G$-set. If $C(G,p)$ denotes the group of
superclass functions defined on the $p$-subgroups of $G$, there are natural
generators $\omega_X$ of $C(G,p)$, and we prove the existence of a well-defined
group homomorphism $\Psi_G:C(G,p)\to D^\Omega(G)$ that sends $\omega_X$ to
$\Omega_X$. The main theorem of the paper is the verification that the subgroup
of $C(G,p)$ consisting of the dimension functions of $k$-orientable real
representations of $G$ lies in the kernel of $\Psi_G$.
