Time series analysis is a widespread task in Natural Sciences, Social Sciences and Engineering. A fundamental problem is finding an expressive yet efficient-to-compute representation of the input time series to use as a starting point to perform arbitrary downstream tasks. In this paper, we build upon recent work using the signature of a path as a feature map and investigate a computationally efficient technique to approximate these features based on linear random projections. We present several theoretical results to justify our approach, we analyze and showcase its empirical performance on the task of learning a mapping between the input controls of a Stochastic Differential Equation (SDE) and its corresponding solution. Our results show that the representational power of the proposed random features allows to efficiently learn the aforementioned mapping.