Gravitation by Charles Misner, Kip Throne, John Wheeler where on chapter 13 page 312 Exercise 13.2 it stated that $$ds^2 =-(1-2M/r) dv^2 + 2dvdr+r(d\theta^2 +\sin^2\theta d\phi^2)$$ (the usual First fundamental form) was an old style notation where the metric had new style notation $$ds^2 =-(1-2M/r) dv\otimes dv + dv\otimes dr +dr\otimes dv +r(d\theta\otimes d\theta +\sin^2\theta d\phi \otimes d\phi)$$
What's the difference between those two notation? Especially was $2dvdr \Rightarrow dv\otimes dr +dr\otimes dv$ instead of $2 dv\otimes dr$ purely for the symmetry of the metric, or does it has some more founding reasons, i.e. the value of $[dv,dr]$ and the corrections?