Orthogonal group is conjugacy-closed in general linear group over reals

Statement
The fact about::orthogonal group over reals $$O(n,\R)$$ over the field of real numbers, is a conjugacy-closed subgroup in the fact about::general linear group over reals $$GL(n,\R)$$. In other words, if two orthogonal matrices are conjugate in the general linear group, they are also conjugate in the orthogonal group.

Note that the statement is not true if we replace the orthogonal group by the fact about::special orthogonal group over reals.