This is almost identical (even one may say "duplicate" of) to this relation between the group $O(3)$ and $SU(2)$. But i would like to ask one more question. Does this relations (given at the "link") mean we can find a isomorphism between $SU(2)$ and $O(3)$ but not continuous group isoomorphism (lie group isomorphism).
Continuos group isomorphism doesn't exist it is clear as mentioned in the "link". But both groups "look like" $SO(3)\otimes \{1,-1\}$ so they should be isomorphic??
Thank you for your answers.
No, they are no isomorphic. For instance, $SU(2)$ only has an element with order $2$ (which is $-\operatorname{Id}$), where is each matrix$$\begin{bmatrix}\pm1&0&0\\0&\pm1&0\\0&0&\pm1\end{bmatrix}$$other than the identity matrix is an element of $O(3)$ which has order $2$.