We basically never say the sixth of November. It sounds positively ancient.
When is your independence day, again?
Anyway, in Australia (and, I suspect, other places that use DD/MM/YYYY) we use “{ordinal} of {month}” (11th of August), “{ordinal} {month}” (11th August), and “{month} {ordinal}” (August 11th) pretty much interchangeably. In writing but not in speaking, we also sometimes use “{number} {month}” (11 August). That doesn’t have any bearing on how we write it short form though, because those are different things. It’s not the defence many Americans seem to think it is of their insane method of writing the short form.