"zu Hause" and "nach Hause" are both idiomatic expressions and belong to every-day language. You can also hear: "zu Haus" and "nach Haus" when the speaker wants to be short.
Otherwise the "e" in Datives is almost extinct, exept in poetic language. "Dem Manne kann geholfen werden" is correct, but it sounds like part of a poem, in every day language you would say "Dem Mann".
Same with Haus. "Überm Hause, da fliegen die Wolken etc." - nice poetic stuff. But the every-day-language version is: "Guck mal! Über dem Haus fliegt ein Hubschrauber!" (Look! There is a helicopter above the house!)
I offered a translation of "we meet the teachers at the house." I was cautioned that "die Lehrer" is plural of "der Lehrer", thanks, and then it returned "we are meeting the teachers at home" underlining "meeting the." So, is it the fact I wrote "at the house" vs "at home"?