The generators for the isospin symmetry are given by

$$
T_{+}=\left|\uparrow\rangle\langle\downarrow\right|,
 \qquad
 T_{-}=\left|\downarrow\rangle\langle\uparrow\right|, 
\qquad 
T_{3}=\frac{1}{2}\left(\left|\uparrow\rangle\langle\uparrow\right|-\left|\downarrow\rangle\langle\downarrow\right|\right),$$

where $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$ form a $2$-dimensional basis of states.

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In the $2$-dimensional basis of states $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$, the generators $T_+$, $T_-$ and $T_3$ can be written as

$T_{+}=\begin{bmatrix}
    0 & 1\\
    0 & 0\\
\end{bmatrix},\qquad T_{+}=\begin{bmatrix}
    0 & 0\\
    1 & 0\\
\end{bmatrix},\qquad T_{3}=\begin{bmatrix}
    1/2 & 0\\
    0 & -1/2\\
\end{bmatrix}.$

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 1. The generators $T_+$, $T_-$ and $T_3$ obey the $SU(2)$ algebra $[T_{i},T_{j}]=i\epsilon_{ijk}T_{k}$ (with $\epsilon_{+-3}=1$). However, I do not get the correct commutation relations using the matrix representations of the generators.
 2. Is it possible to rewrite the generators $T_+$, $T_-$ and $T_3$ in terms of an arbitrary number of states?