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You got a vector $x \neq 0$ with $x \in \mathbb{R}^3$. I have to show that there is a bijection between the stabilizer $\{A \in O_3(\mathbb{R}) \mid Ax = x \}$ and $O_2(\mathbb{R})$. Can someone tell me what the stabilizer looks like and how he is related with the $O_2$ group ? (The group action is the standard matrixproduct as you see)

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HINT: If you think of $\Bbb R^2$ as the $xy$-plane in $\Bbb R^3$, the stabilizer of $\langle 0,0,1\rangle$ is $O_2(\Bbb R)$. –  Brian M. Scott Sep 30 '12 at 10:27
    
thanks. This helps a lot :) –  André Sep 30 '12 at 10:42

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