Source: /cirosantilli/the-orthogonal-group-is-the-group-of-all-matrices-that-preserve-the-dot-product

= The orthogonal group is the group of all matrices that preserve the dot product

When viewed as matrices, it is the group of all matrices that preserve the <dot product>, i.e.:
$$
O(n) = { O \in M(n) | \forall x, y, x^Ty = (Ox)^T (Oy) }
$$
This implies that it also preserves important geometric notions such as <norm (mathematics)> (intuitively: distance between two points) and <angles>.

This is perhaps the best "default definition".