Turbulence in a freely suspended soap film is created by electromagnetic forcing and measured by particle tracking. The velocity fluctuations are shown to be adequately described by the forced Navier-Stokes equation for an incompressible two-dimensional fluid with a linear drag term to model the frictional coupling to the surrounding air. Using this equation, the energy dissipation rates due to air friction and the film's internal viscosity are measured, as is the rate of energy injection from the electromagnetic forcing. Comparison of these rates demonstrates that the air friction is a significant energy dissipation mechanism in the system.