Of course, I'm assuming you are o.k. with skills (and, worse, social skills) in the first place; else, the discussion has no meaning. I like skills, although I take a pretty minimalist approach, using the bare minimum of skills I need to portray the archetypes I want in my games (big post on that coming soon, unless I found out it was already written by someone else).
Intimidation is a recurring skill in WotC D&D, and (IIRC) was a proficiency even before that. Its existence makes some sense, as far as archetypes go: you can easily think of character types that are intimidating despite having no other social skills (for example, ASOIF's The Mountain) - hence, the need for a specific skill.
One problem with this skill is that it is commonly based on Charisma (or a similar stat), making characters that are suave, diplomatic or empathetic good at intimidation (and vice-versa - being the nest at intimidation requires a charismatic type), which is pretty much the opposite of many RPG archetypes; the brute, the bloodthirsty tyrant, the unrelenting fanatic, can all be very intimidating without having any social graces.
One common solution is basing it on Strength, so the character with mighty thews is the most intimidating foe. Again, it seems to make sense, but is he really? Is the dumb brute, by definition, more intimidating than the skilled martial artist or, say, the guy with a gun? Will he be able to intimidate the king with 10 bodyguards between them?
Anyone writing about the subject can tell you intimidation needs something to intimidate with; if you're severely outnumbered or outgunned in one form or another, being intimidated is not that different from plain common sense.
In many cases, intimidation is nothing more that an extreme and violent form of negotiation: "Give me your purse, or I will cut your throat", "surrender and I'll spare your troops", etc. There needn't be a skill roll involved, just a decision about the costs and risks one is willing to take. Charisma and Strength may play a part, sure, but reputation, status, martial skill, and specially the circumstances will be more important than the specific ability of intimidating people.
Having "intimidation" as a skill may encourage players to look at the character sheet instead of focusing on this negotiation (perhaps the same can be said of all social skills, but I don't think so; read on).
What about empty threats? There should certainly be a skill to allow you to intimidate people when you have no real leverage?
Well, there is - it is called "deception", "bluff", or something similar. The sweet-talking character archetype can paint itself as your friend and your foe at the same time, intimidating and seducing at the same time.
Basically, intimidation is about getting leverage, or faking it. The first is probably better handled through role-playing and common sense, the second makes more sense as a skill (since "lying convincingly" is probably a skill in real life too). Different archetypes, such as the brute and the sweet-talker, can achieve the same result through different means, which is why using a single skill might be a bad idea.
Still, I can see some cool uses to intimidation, specially if you want it to play a part in combat; but maybe intimidating a foe during combat to force him to make a mistake is a part of combat skill (or a fighting technique, akin to taunting, etc), not a separate thing that the sweet-talker does better than the seasoned fighter; and in any case I will probably take morale rules and common sense over this.
By the way, e same reasoning can apply to diplomacy: most diplomacy will be some form of etiquette, and thus be covered by "lore", "history" or familiarity with the culture you're dealing it; or be some kind of negotiation, involving common sense (reputation, circumstances, etc) or deception.
In any case, Courtney Campbell was written about the subject here; his whole skill deconstruction series (including one on intimidation) is a big inspiration and well worth a read.