The Skewed Kalman Filter is a powerful tool for statistical inference of asymmetrically distributed time series data. However, the need to evaluate Gaussian cumulative distribution functions (cdf) of increasing dimensions, creates a numerical barrier such that the filter is usually applicable for univariate models and under simplifying conditions only. Based on the intuition of how skewness propagates through the state-space system, a computationally efficient algorithm is proposed to prune the overall skewness dimension by discarding elements in the cdfs that do not distort the symmetry up to a pre-specified numerical threshold. Accuracy and efficiency of this Pruned Skewed Kalman Filter for general multivariate state-space models are illustrated through an extensive simulation study. The Skewed Kalman Smoother and its pruned implementation are also derived. Applicability is demonstrated by estimating a multivariate dynamic Nelson-Siegel term structure model of the US yield curve with Maximum Likelihood methods. We find that the data clearly favors a skewed distribution for the innovations to the latent level, slope and curvature factors.