We report experimental studies of electron-impact ionization of nitrogen (${\mathrm{N}}_{2}$) and oxygen (${\mathrm{O}}_{2}$) molecules using a cold target recoil ion momentum spectrometer (COLTRIMS, reaction microscope). The recoil ion is detected in coincidence with one outgoing electron such that the momentum vectors and consequently the kinetic energies for these final-state particles are determined. The ionization cross sections for producing the doubly and singly charged parent ions were measured as a function of the incident electron energy ranging from 50 to 600 eV. For molecular dications, e.g., ${\mathrm{O}}_{2}^{2+}$, the cross sections are obtained by measuring both the ion time of flight and two-dimensional position spectra, which is demonstrated to be an efficient way to suppress the contribution of the dissociation channels with the ${\mathrm{O}}^{+}$ products as both ions have the identical mass-to-charge ratios. The projectile energy-loss spectra correlated to the final-state ions are obtained, which provide direct evidence of the ionization mechanisms of ${\mathrm{N}}_{2}$ and ${\mathrm{O}}_{2}$ molecules. It is found that the ${\mathrm{O}}_{2}^{2+}$ dication is produced mainly by Auger process after single ionization in $2{\ensuremath{\sigma}}_{g}$ inner-valence shell of ${\mathrm{O}}_{2}$ molecule, while the ${\mathrm{N}}_{2}^{2+}$ dication is generated by the direct removal of two electrons from the outermost $3{\ensuremath{\sigma}}_{g}$ orbital of ${\mathrm{N}}_{2}$ molecule.