  Rotating small AdS black holes exhibit the superradiant instability to
low-frequency scalar perturbations, which is amenable to a complete analytic
description in four dimensions. In this paper, we extend this description to
all higher dimensions, focusing on slowly rotating charged AdS black holes with
a single angular momentum. We divide the spacetime of these black holes into
the near-horizon and far regions and find solutions to the scalar wave equation
in each of these regions. Next, we perform the matching of these solutions in
the overlap between the regions, by employing the idea that the orbital quantum
number $ \ell $ can be thought of as an approximate integer. Thus, we obtain
the complete low-frequency solution that allows us to calculate the complex
frequency spectrum of quasinormal modes, whose imaginary part is determined by
a small damping parameter. Finally, we find a remarkably instructive expression
for the damping parameter, which appears to be a complex quantity in general.
We show that the real part of the damping parameter can be used to give a {\it
universal} analytic description of the superradiant instability for slowly
rotating charged AdS black holes in all spacetime dimensions.
