How to find a real orthogonal matrix of determinant $1$?

Question

A real orthogonal matrix $A$ is proper if $\det A=1 $.

Find $2\times 2$ proper matrix $A$

I tried to use the fact that product of $A$ and its transpose is equal to identity.

But, there were bunch of equations which seem not related to each other and can not find such $A$.

Answer 1

For any $\theta$, $$ \begin{pmatrix} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{pmatrix} $$ is such a matrix. Actually, they all have this form.