The ability of using scattering-type near-field scanning optical microscopy (s-NSOM) to characterize amplitude and phase of optical near fields was investigated. We employ numerical simulations to compute signals scattered by the tip, using a bowtie nano-aperture as the example, and compare with the data obtained from s-NSOM measurements. Through demodulation of higher order harmonic signals, we show that, with the increasing order of harmonic signals, both the simulated and measured near fields are in closer agreement with the anticipated near field results. The polarization-resolved detection also helps to establish a tip-dependent transfer matrix that relates the local field components with the s-NSOM signals, which characterizes the scattering of the tip with respect to different field components. This work illustrates the importance of using higher order signals in obtaining near field in an s-NSOM measurement.