= Solution
Realize the defining <sl2 Lie algebra> <Lie algebra representation> as $\mathbb Cx\oplus\mathbb Cy$, with $H=x\partial_x-y\partial_y$, $E=x\partial_y$, $F=y\partial_x$. A <basis> of the <symmetric power> is $e_i=x^{4-i}y^i$ for $0\leq i\leq4$, with $He_i=(4-2i)e_i$. Hence the <exterior square> <basis> $w_{ij}=e_i\wedge e_j$, $i<j$, has <weight> $8-2(i+j)$.
Collecting equal <weights> gives the complete <weight-space decomposition>:
$$
\begin{array}{c|l|c}
\text{weight}&\text{basis}&\text{multiplicity}\\ \hline
6&w_{01}&1\\
4&w_{02}&1\\
2&w_{03},w_{12}&2\\
0&w_{04},w_{13}&2\\
-2&w_{14},w_{23}&2\\
-4&w_{24}&1\\
-6&w_{34}&1
\end{array}
$$
Every one of the ten exterior <basis vectors> appears exactly once. Thus \b[$U=\bigoplus_{m=6,4,2,0,-2,-4,-6}U_m$], with the displayed <bases> and <dimensions>.
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