I know that $SO(3)$ is a matrix Lie group and therefore a smooth manifold. It has dimension three and I know that it can be, as a manifold constructed by identifying the antipodal points of $\mathbb S^3$.
I also know the "plate trick", that gives a hint that $SO(3)$ is not simply connected. In particular, we cannot take a closed path and shrink it down to a single point. But the plate trick shows us that we can shrink a closed path that goes around twice down to a single point.
I cannot wrap my head around how to imagine the structure of $SO(3)$ as a manifold. Is there maybe a nice projection into the $\mathbb R^4$ or even $\mathbb R^3$, where we can easily see that a closed path that goes around twice can be shrunken down to a single point?