Event-stream representation is the first step for many computer vision tasks using event cameras. It converts the asynchronous event-streams into a formatted structure so that conventional machine learning models can be applied easily. However, most of the state-of-the-art event-stream representations are manually designed and the quality of these representations cannot be guaranteed due to the noisy nature of event-streams. In this paper, we introduce a data-driven approach aiming at enhancing the quality of event-stream representations. Our approach commences with the introduction of a new event-stream representation based on spatial-temporal statistics, denoted as EvRep. Subsequently, we theoretically derive the intrinsic relationship between asynchronous event-streams and synchronous video frames. Building upon this theoretical relationship, we train a representation generator, RepGen, in a self-supervised learning manner accepting EvRep as input. Finally, the event-streams are converted to high-quality representations, termed as EvRepSL, by going through the learned RepGen (without the need of fine-tuning or retraining). Our methodology is rigorously validated through extensive evaluations on a variety of mainstream event-based classification and optical flow datasets (captured with various types of event cameras). The experimental results highlight not only our approach's superior performance over existing event-stream representations but also its versatility, being agnostic to different event cameras and tasks.