Recently, S. Mitryukovskiy et al. presented experimental evidence showing that backward Amplified Spontaneous Emission (ASE) at 337 nm can be obtained from plasma filaments in nitrogen gas pumped by circularly polarized 800 nm femtosecond pulses (Opt. Express, 22, 12750 (2014)). Here, we report that a seed pulse injected in the backward direction can be amplified by ~200 times inside this plasma amplifier. The amplified 337 nm radiation can be either linearly or circularly polarized, dictated by the seeding pulse, which is distinct from the non-polarized nature of the ASE. We performed comprehensive measurements of the spatial profile, optical gain dynamics, and seed pulse energy dependence of this amplification process. These measurements allow us to deduce the pulse duration of the ASE and the amplified 337 nm radiation as well as the corresponding laser intensity inside the plasma amplifier. It indicates that the amplification is largely in the unsaturated regime and that further improvement of laser energy is possible. Moreover, we observed optical gain in plasma created in ambient air. This represents an important step towards future applications exploiting backward lasing for remote atmospheric sensing.