We study the composition of spectator matter in ultracentral collisions of heavy relativistic nuclei with neutron-rich periphery known as a neutron skin (NS). The yields of spectator neutrons and protons in ultracentral $^{208}$Pb–$^{208}$Pb collisions at the LHC were calculated within a new version of Abrasion-Ablation Monte Carlo for Colliders (AAMCC) model. Calculations demonstrate variations of the cross sections $\sigma(N_n, N_p)$ to produce given numbers of spectator neutrons $N_n$ and protons $N_p$ depending on the parameters of NS in $^{208}$Pb, in particular, for events with low $N_n$. The relative difference between $\sigma(N_n, N_p)$ obtained with different NS thickness reaches 150%. Furthermore, the values of $\sigma(N_n, N_p)$ calculated with same difference between neutron and proton RMS-radii, but with different neutron half-density radius and diffuseness parameter, differ by 45%. The considered cross sections can be measured in the ALICE experiment at the LHC for $^{208}$Pb–$^{208}$Pb to restrict the parameters of NS in colliding nuclei.