Open relation extraction (ORE) aims to assign semantic relationships among arguments, essential to the automatic construction of knowledge graphs (KG). The previous ORE methods and some benchmark datasets consider a relation between two arguments as definitely existing and in a simple single-span form, neglecting possible non-existent relationships and flexible, expressive multi-span relations. However, detecting non-existent relations is necessary for a pipelined information extraction system (first performing named entity recognition then relation extraction), and multi-span relationships contribute to the diversity of connections in KGs. To fulfill the practical demands of ORE, we design a novel Query-based Multi-head Open Relation Extractor (QuORE) to extract single/multi-span relations and detect non-existent relationships effectively. Moreover, we re-construct some public datasets covering English and Chinese to derive augmented and multi-span relation tuples. Extensive experiment results show that our method outperforms the state-of-the-art ORE model LOREM in the extraction of existing single/multi-span relations and the overall performances on four datasets with non-existent relationships. Our code and data are publicly available.