  Coupled oscillator models where $N$ oscillators are identical and
symmetrically coupled to all others with full permutation symmetry $S_N$ are
found in a variety of applications. Much, but not all, work on phase
descriptions of such systems consider the special case of pairwise coupling
between oscillators. In this paper, we show this is restrictive - and we
characterise generic multi-way interactions between oscillators that are
typically present, except at the very lowest order near a Hopf bifurcation
where the oscillations emerge. We examine a network of identical weakly coupled
dynamical systems that are close to a supercritical Hopf bifurcation by
considering two parameters, $\epsilon$ (the strength of coupling) and $\lambda$
(an unfolding parameter for the Hopf bifurcation). For small enough $\lambda>0$
there is an attractor that is the product of $N$ stable limit cycles; this
persists as a normally hyperbolic invariant torus for sufficiently small
$\epsilon>0$. Using equivariant normal form theory, we derive a generic normal
form for a system of coupled phase oscillators with $S_N$ symmetry. For fixed
$N$ and taking the limit $0<\epsilon\ll\lambda\ll 1$, we show that the
attracting dynamics of the system on the torus can be well approximated by a
coupled phase oscillator system that, to lowest order, is the well-known
Kuramoto-Sakaguchi system of coupled oscillators. The next order of
approximation genericlly includes terms with up to four interacting phases,
regardless of $N$. Using a normalization that maintains nontrivial interactions
in the limit $N\rightarrow \infty$, we show that the additional terms can lead
to new phenomena in terms of coexistence of two-cluster states with the same
phase difference but different cluster size.
