Aspect Sentiment Triplet Extraction (ASTE) aims to extract aspect term, sentiment and opinion term triplets from sentences and tries to provide a complete solution for aspect-based sentiment analysis (ABSA). However, some triplets extracted by ASTE are confusing, since the sentiment in a triplet extracted by ASTE is the sentiment that the sentence expresses toward the aspect term rather than the sentiment of the aspect term and opinion term pair. In this paper, we introduce a more fine-grained Aspect-Sentiment-Opinion Triplet Extraction (ASOTE) Task. ASOTE also extracts aspect term, sentiment and opinion term triplets. However, the sentiment in a triplet extracted by ASOTE is the sentiment of the aspect term and opinion term pair. We build four datasets for ASOTE based on several popular ABSA benchmarks. We propose a Position-aware BERT-based Framework (PBF) to address this task. PBF first extracts aspect terms from sentences. For each extracted aspect term, PBF first generates aspect term-specific sentence representations considering both the meaning and the position of the aspect term, then extracts associated opinion terms and predicts the sentiments of the aspect term and opinion term pairs based on the sentence representations. Experimental results on the four datasets show the effectiveness of PBF.