We propose and experimentally demonstrate an optical reservoir computing system in free space, using second-harmonic generation for nonlinear kernel functions and a scattering medium to enhance reservoir nodes interconnection. We test it for one-step and multi-step predication of Mackey-Glass time series with different input-mapping methods on a spatial light modulator. For one-step prediction, we achieve 1.8 × 10-3 normalized mean squared error (NMSE). For the multi-step prediction, we explore two different mapping methods: linear-combination and concatenation, achieving 16-step prediction with NMSE as low as 3.5 × 10-4. Robust and superior for multi-step prediction, our approach and design have potential for parallel data processing tasks such as video prediction, speech translation, and so on.