All spells need a clear path to the target unless they state otherwise
5e has eschewed the term "line of effect" in favor of a stipulation that there must be a "clear path to the target" as outlined in the spell targeting rules:
To target something, you must have a clear path to it, so it can't be behind total cover. (PHB 204)
Enchantment spells are no exception to this and crown of madness has no text that specifically excludes it from the rule, and therefore the restriction applies.
Being in another room almost certainly qualifies as full cover here unless there is a window or passage leading to the room from where you are in the other room.
A target has total cover if it is completely concealed by an obstacle.
However, if there was an opening or some other path in between the rooms you could certainly target the wizard then. However...
This plan would not work anyways because of line of sight
Your plan has another issue: once you start casting another concentration spell any concentration spells end instantly.
As soon as you start casting a spell or using a special ability that requires concentration, your concentration on another effect ends instantly. (XGE)
The reason this matters is because crown of madness requires line of sight as well as line of effect. However, if the wizard is in another room, the only way you can see them is with an ability or spell (in your example arcane eye). However, as soon as you start casting crown of madness, arcane eye will stop working which means you then no longer have line of sight to the wizard. No line of sight means you can't target the wizard with crown of madness.
Also note that range refers to the distance between you (the caster) and your target and arcane eye isn't going to help close that range.