We describe a relation between the symmetry energy coefficients c(sym)(rho) of nuclear matter and a(sym)(A) of finite nuclei that accommodates other correlations of nuclear properties with the low-density behavior of c(sym)(rho). Here, we take advantage of this relation to explore the prospects for constraining c(sym)(rho) of systematic measurements of neutron skin sizes across the mass table, using as example present data from antiprotonic atoms. The found constraints from neutron skins are in harmony with the recent determinations from reactions and giant resonances.