An R-matrix calculation for electron-helium scattering is described in which the eleven lowest target states of helium are included, using configuration interaction wavefunctions. Excitation cross sections are calculated up to 30 eV from the 1¹S ground state and the metastable 2³S and 2¹S states of helium to the higher n=2 states, and, for the first time in an ab initio calculation, to the n=3 states. The authors confirm their preliminary finding that the explicit coupling of the n=3 channels significantly reduces the cross sections calculated in an earlier five-state approximation, by up to 50% for n=1 to n=2 transitions at energies above the n=3 thresholds. Using the new cross section data, rate coefficients are calculated for temperatures up to 50000K for the transitions considered.