Embed $\mathbf{S}^2$ in $\mathbf{R}^3$. For each rotation has a unique vector line on which it acts as the identity, and acts as a plane rotation on the orthogonal plane to this vector line, take both such vector lines for your two rotations of same angle, and take both associated orthogonal planes. As both rotations have same angle, your two rotations have the same matrix in two (different, associated to the decompostion "fixed line$\oplus$orthogonal plane") orthonormal basis of $\mathbf{R}^3$. Then simply take the change of basis matrix for this couple of basis, and it will give you the matrix realizing conjugation between you two rotations, and ensure that this matrix is in $SO(3,\mathbf{R})$, which is always possible.