Preliminaries
Let $V = \mathbb R^n$, and $W = S_n$ the symmetric group. We consider the set $\Phi = \{±(e_i - e_j) \mid 1 ≤ i < j ≤ n\} $ (where the $e_i$ are the standard vectors), which is a root system in $V$. We let $S_n$ act on $\mathbb R^n$ by permuting the standard vectors $e_1, \dots, e_n$ (permuting the subscripts). We have that $\langle W(\Phi) \rangle := \{s_\alpha: \alpha \in \Phi \} = S_n $ (where $s_\alpha$ is the reflection that sends $\alpha$ to $-\alpha$ and fixes the hyperplane orthogonal to $\alpha$); in other words, the reflections of $\Phi$ generate $S_n$. (The $s_{\alpha}$ for $\alpha = e_i - e_j$ are exactly the transpositions, and it's commonly known that $S_n$ is generated by transpositions.) In this context, $\Phi$ is generally considered the standard root system for $S_n$.
Let $\Delta$ be a simple system of $\Phi$, and $\Pi$ the respective positive system for $\Delta$. I already know that each $w \in W$ can be written as a product of positive/simple reflections. For $w \in W$, we call $l(w)$ the length of $W$ if $l(w) = r \in \mathbb N$ and $r$ is the minimal number so that there exists $\alpha_1, \dots, \alpha_r \in \Delta $ with $w = s_{\alpha_1} \dots s_{\alpha_r}$.
It can be shown that the length of a $w$ is exactly the number of positive roots (for some fixed $\Delta$ and $\Pi$) that are sent to negative roots by $w$; in other words, $l(w) = |\Pi \cap w^{-1}(-\Pi)|$. There are some more interesting properties of the length function, like $l(w w') \equiv l(w) + l(w') \mod 2$ or $l(w s_\alpha) = l(w) ± 1$, but in general, the length behaves "as one would expect".
We call a $w_0 \in W$ the longest element if $l(w_0)$ is maximal; that is, if $l(w) ≤ l(w_0) \quad \forall w ≠ w_0$. It can be shown that the longest element is unique (so we have $l(w) < l(w_0) \forall w ≠ w_0$), and that for any $w \in W$, we can get to $w_0$ by multiplying $w$ with all $s_\alpha$ for which the length increases. Especially, we have $l(w_0) = |\Pi|$ (because $w_0$ must send all positive roots to negative ones, which was one of the characterizations of the length function).
My question
I want to find the longest element $w_0$ of $S_n$ for the root system $\Phi = \{±(e_i - e_j) \mid 1 ≤ i < j ≤ n\}$ and the positive system $\Pi = \{(e_i - e_j) \mid 1 ≤ i < j ≤ n\}$ (by letting $S_n$ act on $e_1, \dots, e_n$ by permuting the coordinates, as described above).
Now I've already stated above what this $w_0$ must satisfy: it must be an element of $S_n$, that, acting on $\Phi$ in the way described above, has maximal length $|\Pi|$. I also know that all the $e_i - e_j$ must be sent to $e_j - e_i$ because $w_0$ sends all positive roots to negative ones. I'm not really sure how to go from there though; I'm still lacking the intuition on how to get from the roots back to the elements of $S_n$.
I've also already stated above that $w_0$ can be gotten by suczessively multiplying the $s_\alpha, \alpha \in \Delta$ together for all the $\alpha$ for which the length increases, but my problem here is that I'm not entirely sure what the respective simple system $\Delta$ for the given positive system $\Phi$ looks like.
I might be missing something obvious here, but I've been sitting on this question for a while now and I'm out of ideas.
The longest element for each type, in particular for type $\mathsf{A}_n$, is described in the tables at the end of Bourbaki's "Lie groups and Lie algebras", Chapters 4-6. - It is decribed through its action on the simple roots, or which is the same, through its action on the nodes of the Dynkin diagram. In this way, $w_o$ corresponds to a diagram automorphism.
For type $\mathsf{A}_n$, the element $w_o$ corresponds to the diagram automorphism which swaps the line in the middle. In formulas, we have $w_o(\alpha_i)=-\alpha_{n+1-i}$ for all $1\leq i\leq n$, where $\alpha_i$ is the $i$th simple root, i.e. we have $\alpha_i=\epsilon_i-\epsilon_{i+1}$.
I am not sure what kind of description of $w_o$ you are looking for. But from this, you can figure out the action of $w_o$ on each $\epsilon_i$, and even describe $w_o$ as a permutation.