Channel knowledge map (CKM) is viewed as a digital twin of wireless channels, providing location-specific channel knowledge for environment-aware communications. A fundamental problem in CKM-assisted communications is how to construct the CKM efficiently. Current research focuses on interpolating or predicting channel knowledge based on error-free channel knowledge from measured regions, ignoring the extraction of channel knowledge. This paper addresses this gap by unifying the extraction and representation of channel knowledge. We propose a novel CKM construction framework that leverages the received signals of the base station (BS) as online and low-cost data. Specifically, we partition the BS coverage area into spatial grids. The channel knowledge per grid is represented by a set of multi-path powers, delays, and angles, based on the principle of spatial consistency. In the extraction of these channel parameters, the challenges lie in strong inter-cell interferences and non-linear relationship between received signals and channel parameters. To address these issues, we formulate the problem of CKM construction into a problem of Bayesian inference, employing a block-sparsity prior model to characterize the path-loss differences of interferers. Under the Bayesian inference framework, we develop a hybrid message-passing algorithm for the interference-cancellation-based CKM construction. Based on the CKM, we obtain the joint frequency-space covariance of user channel and design a CKM-assisted Bayesian channel estimator. The computational complexity of the channel estimator is substantially reduced by exploiting the CKM-derived covariance structure. Numerical results show that the proposed CKM provides accurate channel parameters at low signal-to-interference-plus-noise ratio (SINR) and that the CKM-assisted channel estimator significantly outperforms state-of-the-art counterparts.