Density matrices describing H(n=3) atoms produced in collisions of 20- to 100-keV protons with He atoms have been determined experimentally. In the experiment the intensity and polarization of Balmer-α radiation emitted from a He gas cell are measured as a function of the strength of an externally applied electric field. Electric fields are applied in a direction either axial to or transverse to the proton beam. Density matrices are extracted by detailed analysis of the optical data. Data are obtained for each field direction and then analyzed, separately and in combination, to yield density matrices. Satisfactory agreement is found between density matrices determined from axial and transverse electric field data except at the lowest energies studied. Some nonzero density-matrix elements are determined more accurately using axial electric fields than with transverse fields, while other elements are more accurately determined using transverse electric fields. The combined analysis using data from both field directions gives a better determination of the density matrix than the separate data sets. Results for the H(n=3) electron-transfer cross sections (relative to 3s), the electric dipole moment of the charge distribution 〈d〉z, a first-order moment of the current distribution 〈L×A〉z,s, and the average coherence Tr(σ32) are obtained. The experimental results are compared to two recent calculations using the augmented atomic orbital (AO+) theory and the continuum distorted-wave approximation with post-collision interaction theory, and to one recent experimental measurement of the diagonal density-matrix elements. Both theories show qualitative agreement with the general trends in the data. The AO+ method gives better quantitative agreement. The experimental results are displayed in graphical form as distributions of the electronic charge D(r) and of the electronic current density j(r).