Context. In order to allow for a comparison with the measurements from other antenna systems, the voltage power spectral density measured by the Radio and Plasma waves receiver (RPW) on board Solar Orbiter needs to be converted into physical quantities that depend on the intrinsic properties of the radiation itself (e.g., the brightness of the source). Aims. The main goal of this study is to perform a calibration of the RPW dipole antenna system that allows for the conversion of the voltage power spectral density measured at the receiver's input into the incoming flux density. Methods. We used space observations from the Thermal Noise Receiver (TNR) and the High Frequency Receiver (HFR) to perform the calibration of the RPW dipole antenna system. Observations of type III bursts by the Wind spacecraft are used to obtain a reference radio flux density for cross-calibrating the RPW dipole antennas. The analysis of a large sample of HFR observations (over about ten months), carried out jointly with an analysis of TNR-HFR data and prior to the antennas' deployment, allowed us to estimate the reference system noise of the TNR-HFR receivers. Results. We obtained the effective length, l(eff), of the RPW dipoles and the reference system noise of TNR-HFR in space, where the antennas and pre-amplifiers are embedded in the solar wind plasma. The obtained l(eff) values are in agreement with the simulation and measurements performed on the ground. By investigating the radio flux intensities of 35 type III bursts simultaneously observed by Wind and Solar Orbiter, we found that while the scaling of the decay time as a function of the frequency is the same for the Waves and RPW instruments, their median values are higher for the former. This provides the first observational evidence that Type III radio waves still undergo density scattering, even when they propagate from the source, in a medium with a plasma frequency that is well below their own emission frequency.