Ciro Santilli $£ Sponsor €¥ 中国独裁统治 China Dictatorship 新疆改造中心、六四事件、法轮功、郝海东、709大抓捕、2015巴拿马文件 邓家贵、低端人口、西藏骚乱
When viewed as matrices, it is the group of all matrices that preserve the dot product, i.e.:
This implies that it also preserves important geometric notions such as norm (intuitively: distance between two points) and angles.
This is perhaps the best "default definition".