Let $\beta_n$ denote the flag $h$-vector (as defined in EC1, Section
3.13) of the partition lattice $\Pi_n$ (EC1, Example 3.10.4). Then
$$ \mathrm{mag}_{n,{n-1\choose 2}-j} = \sum_S \beta_n(S), $$
where $S$ ranges over all subsets of $\lbrace 1,2,\dots,n-2\rbrace$ whose
elements sum to $j$. An explicit formula for $\beta_n(S)$ is given by
$$ \beta_n(S) = \sum_{T\subseteq S} (-1)^{|S-T|}
\alpha_n(T), $$
where if the elements of $T$ are $t_1<\cdots < t_k$, then
$$ \alpha_n(T) = S(n,n-t_1)S(n-t_1,n-t_2)
S(n-t_2,n-t_3)\cdots S(n-t_{k-1},n-t_k). $$
Here $S(m,j)$ denotes a Stirling number of the second kind.
Addendum. A combinatorial description of the mag numbers is
somewhat complicated. Consider all ways to start with the $n$ sets
$\lbrace 1 \rbrace,\dots, \lbrace n \rbrace$. At each step we take two
of our sets and replace them by their union. After $n-1$ steps we will
have the single set $\lbrace 1,2,\dots,n \rbrace$. An example for
$n=6$ is (writing a set like $\lbrace 2,3,5\rbrace$ as 235)
1-2-3-4-5-6, 1-2-36-4-5, 14-36-2-5, 14-356-2, 14356-2, 123456. At the
$i$th step suppose we take the union of two sets $S$ and $T$. Let
$a_i$ be the least integer $j$ such that $j$ belongs to one of the
sets $S$ or $T$, and some number smaller than $j$ belongs to the other
set. For the example above we get $(a_1,\dots,a_5)=(6,4,5,3,2)$. If
$\nu$ denotes this merging process, then let $f(\nu) = \sum i$,
summed over all $i$ for which $a_i>a_{i+1}$. For the above example,
$f(\nu) = 1+3+4=8$. (The number $f(\nu)$ is called the major index
of the sequence $(a_1,\dots,a_{n-1})$.) Then
$\mathrm{mag}_{n,{n-1\choose 2}-j}$ is the number
of merging processes $\nu$ for which $f(\nu)=j$. This might look
completely contrived to the uninitiated, but it is very natural within
the theory of flag $h$-vectors.
Best Answer
In 3 dimensions, rotations, i.e., transformations corresponding to orthogonal $U$ with determinant 1, are generated by the (orbital) angular momentum operator $\vec{L} $ with components $L_i =-i \epsilon_{ijk} x_j \,\partial / \partial x_k $. By Euler's rotation theorem, any given such transformation can be effected by rotating around a specific axis $\vec{e} $ by an angle $\alpha $. Then, the desired rotation operator is $$ \exp \left(-i \alpha \ \vec{e} \cdot \vec{L} \right) \ . $$ In other than 3 dimensions, there isn't, of course, such an intuitive description in terms of a vector axis and an angle, but the modification is purely on the level of the rotation theorem -- once this is adapted, one will still then generate the rotations using the antisymmetric tensor operator $x_j \,\partial / \partial x_k - x_k \,\partial / \partial x_j $.