Classification of multivariate time series (MTS) has been tackled with a large variety of methodologies and applied to a wide range of scenarios. Reservoir computing (RC) provides efficient tools to generate a vectorial, fixed-size representation of the MTS that can be further processed by standard classifiers. Despite their unrivaled training speed, MTS classifiers based on a standard RC architecture fail to achieve the same accuracy of fully trainable neural networks. In this article, we introduce the reservoir model space, an unsupervised approach based on RC to learn vectorial representations of MTS. Each MTS is encoded within the parameters of a linear model trained to predict a low-dimensional embedding of the reservoir dynamics. Compared with other RC methods, our model space yields better representations and attains comparable computational performance due to an intermediate dimensionality reduction procedure. As a second contribution, we propose a modular RC framework for MTS classification, with an associated open-source Python library. The framework provides different modules to seamlessly implement advanced RC architectures. The architectures are compared with other MTS classifiers, including deep learning models and time series kernels. Results obtained on the benchmark and real-world MTS data sets show that RC classifiers are dramatically faster and, when implemented using our proposed representation, also achieve superior classification accuracy.