The idea is one of many that look a lot better on paper than they do in hard engineering and aerodynamics. All that free oxygen is so appealing...

I take some pride that, over the past thirty-five years, I have had on my drawing boards (or computer) almost every conceivable system for getting into space cheaply. We toyed with scramjet/rocket systems. Turns out that what an orbital launch vehicle really, really wants to do is get out of the goddam atmosphere as soon as possible so it can open up and get some velocity without having to worry about immense drag and an awful heat load. When you get to the higher mach numbers just getting the air sorted out so you can burn it becomes a major issue in itself.

It can be done, and the Skylon concept is workable as a technology. But it's about as useful as trying to make a supersonic submarine.

Hydrogen is a great fuel burned with liquid oxygen, and you are right that you need big-ass tanks to carry it. That's what that giant balloon on the belly of the Shuttle orbiter is for. You have no idea how big it would be if it had to haul that stack off the ground without those solids.

Most staged rockets use liquid oxygen and some hydrocarbon (RP1, kerosene, propane, or something else) in the first stage. Performance is okay for a first stage, and the propellants are nice and dense (relatively). You can get a lot of it in a moderate-sized tank. Upper stages, which use large expansion nozzles and where high performance really pays, sometimes use LOX-hydrogen. The Centaur is one of the most common in the U.S. The SpaceX vehicles use the same propellants for all stages. They lose a bit of performance, but it saves a shitload of development and propellant-handling money.

In recent years, many of the designs I've worked on are air-launched, like Virgin Galactic's designs, only with a lot bigger rockets and planes for true orbital launches. You gain nothing measurable in velocity with an air launch, but it allows you to use a high-altitude engine in a lower-drag environment right off the bat, and you pick up a lot of performance on your first stage.

Edit: sorry, guess I started to ramble on a bit there. Don't get to talk shop much nowadays.