  It is known that a $C_0$-semigroup of Hilbert space operators is
$m$-isometric if and only if its generator satisfies a certain condition, which
we choose to call $m$-skew-symmetry. This paper contains two main results: We
provide a Lumer--Phillips type characterization of generators of $m$-isometric
semigroups. This is based on the simple observation that $m$-isometric
semigroups are quasicontractive. We also characterize cogenerators of
$2$-isometric semigroups. To this end, our main strategy is to construct a
functional model for $2$-isometric semigroups with analytic cogenerators. The
functional model yields numerous simple examples of $2$-isometric semigroups,
but also allows for the construction of a closed, densely defined,
$2$-skew-symmetric operator which is not a semigroup generator.
