In Lec 26, equation (3) involves "antisymmetrizing by u, w (permutting u, w and subtracting from original formula)...

What does antisymmetrizing mean? Or Permutting for that matter? I'm not sure how we move from (2) to (3) to (4)...

Can someone kindly clarify?

It means that we have expression $\mathcal{P}(u,w)$ depending on $u$ and $v$. Now we consider expression $\mathcal{P}(w,u)$ so interchange $u$ and $w$.

"Symmetrizing" would mean that we consider $\mathcal{P}(u,w)+\mathcal{P}(w,u)$ (or rather divided by $2$).

"Antisymmetrizing" would mean that we consider $\mathcal{P}(u,w)-\mathcal{P}(w,u)$ (or rather divided by $2$).