This study presents the approach to aspect-based sentiment analysis where a named entity of certain category is considered as an aspect. Such task formulation is a novelty and opens up the opportunity to determine writers' attitude to organizations and people considered in texts. This task required a dataset of Russian-language sentences where sentiment with respect to certain named entities would be labeled, which we collected using a crowdsourcing platform. Sentiment determination is based on a deep neural network with attention mechanism, and ELMo language model for word vector representation. The proposed model is validated on available data of a similar task. The resulting performance (by the f1-micro metric) on the collected dataset is 0.72, which is the new state of the art for the Russian language.