This work investigates asymptotic behaviors of spatial-sign based location and scatter estimators, i.e., the sample spatial median and its associated spatial-sign covariance matrix, in high-dimensional frameworks. Two stochastic representations for the spatial median are provided with explicit forms, which can characterize the first and second order fluctuations of the spatial median, in the almost sure sense. Beyond this, a new central limit theorem is established for linear spectral statistics of the spatial-sign covariance matrix. All these results are obtained under a general population model that covers the popular independent components model and the family of elliptical distributions.