  A precise and model-independent determination of the neutron distribution
radius $R_{\rm n}$ and thus the neutron skin thickness $R_{\rm skin}$ of atomic
nuclei is of fundamental importance in nuclear physics, particle physics and
astrophysics but remains a big challenge in terrestrial labs. We argue that the
nearby core-collapse supernova (CCSN) in our Galaxy may render a neutrino flux
with unprecedentedly high luminosity, offering perfect opportunity to determine
the $R_{\rm n}$ and $R_{\rm skin}$ through the coherent elastic
neutrino-nucleus scattering (CE$\nu$NS). We evaluate the potential of
determining the $R_{\rm n}$ of lead (Pb) via CE$\nu$NS with the nearby CCSN
neutrinos in the RES-NOVA project which is designed to hunt CCSN neutrinos
using an array of archaeological Pb based cryogenic detectors. We find that an
ultimate precision of $\sim 0.1 \%$ for the $R_{\rm n}$ ($\sim 0.006$ fm for
the $R_{\rm skin}$) of Pb can be achieved via RES-NOVA in the most optimistic
case that the CCSN explosion were to occur at a distance of $\sim 1$ kpc from
the Earth.
