A search is performed for a heavy Majorana neutrino (N), produced by leptonic decay of a W boson propagator and decaying into a W boson and a lepton, with the CMS detector at the LHC. The signature used in this search consists of two same-sign leptons, in any flavor combination of electrons and muons, and at least one jet. The data were collected during 2016 in proton-proton collisions at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 35.9 fb$^{-1}$. The results are found to be consistent with the expected standard model backgrounds. Upper limits are set in the mass range between 20 and 1600 GeV in the context of a Type-I seesaw mechanism, on |$V_\mathrm{eN}$|$^2$, |$V_{\mu\mathrm{N}}$|$^2$, and |$V_{\mathrm{eN}}$$V^*_{\mu\mathrm{N}}$|$^2$ / (|$V_\mathrm{eN}$|$^2$ + |$V_{\mu\mathrm{N}}$|$^2$), where $V_{\ell\mathrm{N}}$ is the matrix element describing the mixing of N with the standard model neutrino of flavor $\ell =$ e, $\mu$. For N masses between 20 and 1600 GeV, the upper limits on |$V_{\ell\mathrm{N}}$|$^2$ range between 2.3 $\times$ 10$^{-5}$ and unity. These are the most restrictive direct limits for heavy Majorana neutrino masses above 430 GeV.