= Solution
The <special unitary group> is $\mathrm{SU}(2)=\{g\in M_2(\mathbb C):g^*g=I,\ \det g=1\}$. The <special orthogonal group> is $\mathrm{SO}(3)=\{R\in M_3(\mathbb R):R^TR=I,\ \det R=1\}$. <Conjugation> $A\mapsto gAg^{-1}$ preserves <trace> and skew-Hermiticity: for unitary $g$, $(gAg^{-1})^*=gA^*g^{-1}=-gAg^{-1}$. It therefore defines the representation on the real space $V$ described in the following parts. The bar over the transpose in the PDF is essential; the condition is $A+A^*=0$, not ordinary antisymmetry over the <complex numbers>.
An <orthonormal basis> for the pairing proved below is
$$
E_1=\frac i{\sqrt2}\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad
E_2=\frac1{\sqrt2}\begin{pmatrix}0&1\\-1&0\end{pmatrix},\quad
E_3=\frac i{\sqrt2}\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
$$
Put $J_j=\sqrt2E_j$, so $J_j^2=-I$. For any unit real vector $v$, $J(v)=\sum_jv_jJ_j$ also squares to $-I$. Thus $g=\cos(\theta/2)I+\sin(\theta/2)J(v)$ belongs to $\mathrm{SU}(2)$. <Conjugation> fixes $J(v)$ and rotates its perpendicular plane by angle $\theta$ up to the orientation convention of this basis, as follows either by multiplying the displayed matrices or from their anticommutator relation $J(v)J(w)+J(w)J(v)=-2(v\cdot w)I$. Since every three-dimensional proper rotation has an axis and an angle, this proves surjectivity onto $\mathrm{SO}(3)$.
If an element is in the <kernel>, it commutes with each $E_j$. Commutation with $E_3$ forces it to be diagonal; commutation with $E_1$ forces its two diagonal entries to agree. Thus it is a scalar matrix, and <determinant> one and unitarity give exactly $\{I,-I\}$. Therefore \b[$\boxed{\mathrm{SO}(3)\cong\mathrm{SU}(2)/\{\pm I\}}$].
The finite-dimensional complex <irreducible representations> of $\mathrm{SU}(2)$ are $\operatorname{Sym}^m(\mathbb C^2)$, $m\ge0$, of dimension $m+1$. The central element $-I$ acts by $(-1)^m$, so precisely the even ones descend. \b[The irreducibles of $\mathrm{SO}(3)$ are $\boxed{\operatorname{Sym}^{2\ell}(\mathbb C^2),\ \ell=0,1,2,\ldots}$], viewed as representations of the quotient, with dimensions $2\ell+1$.
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