I am new to Lie goups and representation theory, and I am confused about irreducible representations of $SU(2)$ of dimension $n>2$.

Let us take the more intuitive example of $SO(2)$. I seems obvious that any faithful 3-dimensional representation is similar to a representation of the form
$$\begin{pmatrix}
\cos \theta & -\sin \theta & 0 \\
\sin \theta & \cos \theta & 0 \\
0 & 0 & 1
\end{pmatrix}.$$
This representation is clearly the direct sum of a 2-dimensional representation of $SO(2)$ and the trivial representation, so any 3-dimensional representation of $SO(2)$ is reducible. Is this basically correct?

It seems to me like the same should be true for $SU(2)$, but I am reading that spin-1 particles transform according to a 3-dimensional representation, that spin-3/2 particles transform according to a 4-dimensional representation; and so on. In fact, [Wikipedia](https://en.wikipedia.org/wiki/Spin_(physics)#Rotations) states that *"There is one n-dimensional irreducible representation of SU(2) for each dimension"*.

How can this be? I.e., how can there be irreducible representations of $SU(2)$ of dimension $n>2$, when two dimensions is "enough"?