What is the orthogonal group $\mathrm{O}(n)$ ? What is the special orthogonal group $\mathrm{SO}(n) ?$

Show that every element of $\mathrm{SO}(3)$ has an eigenvector with eigenvalue $1 .$

Is it true that every element of $\mathrm{O}(3)$ is either a rotation or a reflection? Justify your answer.