You can say "Aujourd'hui on est mardi/or Aujourd'hui c'est mardi". There aren't "ne" between "je" and "aime" because it's when you speak but when you write, you must say "ne"

You can't say "nous sommes le mardi". we don't use article with days of the week, except if we want to say something repete every week. ex : J'ai un rendez-vous lundi. = I have an appointment on Monday. ex : j'ai un cours le lundi. = I have lessons on Mondays (every Mondays) As for the "ne... pas", the good way is to say both all the time, but when we speak fast, we like to drop the "ne". It's a habbit, not a necessity. And in writing, it's bad form to skip the "ne". I hope it explains it all!