We present the inclusive invariant-mass, transverse momentum and rapidity distributions of dielectrons (e$^{+}$e$^{-}$ pairs) in p+p interactions at 3.5 GeV beam kinetic energy. In the vector-meson mass region, a distinct peak corresponding to direct $omega$ decays is reconstructed with 2% mass resolution. The data is compared to predictions from three model calculations. Due to the large acceptance of the HADES apparatus for $e^{+}e^{-}$ invariant masses above 0.2 GeV/$c^{2}$ and for transverse pair momenta p$_{t}$ $ <$ 1 GeV/$c$, acceptance corrections are to a large extent model independent. This allows us to extract from dielectron data for the first time at this energy the inclusive production cross sections for light vector mesons. Inclusive production cross sections for $pi^o$ and $eta$ mesons are also reported. The obtained results will serve as an important reference for the study of vector meson production in proton-nucleus and heavy-ion collisions. Furthermore, using this data, an improved value for the upper bound of the branching ratio for direct $eta$ decays into the electron-positron channel is obtained.
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