One of the provided answers is wrong, at least in the UK (and probably throughout the EU). Under the Data Protection Act you have the right to request any and all data that a company has recorded about you (with some exceptions) and even in some cases, data which is not about you but has some direct effect upon you.
Taking the legal route, you would have to put the request in writing (a "subject access request") and cover the company's admin costs (up to a statutory limit of £10). They have up to 40 days to respond, after which they are breaking the law. It is highly unusual for a company to dispute your identity for SARs, but attaching a copy of your passport should lay any such concerns to rest.
It's not necessary to take the legal route however. Virtually every company you contact by telephone in the UK will provide any information they have to hand about you, so long as you can convince them that you are that person. This will usually consist of two or three simple "data protection" questions, such as asking you to verify your address, date of birth, account number, etc. In the context of an airline they might ask you for your booking reference number or some such similar information. You probably don't have that info.
Even if you are unable to answer all of the data protection questions, that isn't the end of your options. Most companies will have a process for resolving queries where a customer forgets or cannot locate certain details. This might involve emailing a copy of your passport, for example. If it were me, I'd simply call up the airline and tell them the truth - "someone else has booked a ticket; I don't know the destination or booking reference, but it's in my name and I would like to know the details".
The fact that you do not know the airline is not an issue, because you can simply contact them all. Yes, it is time consuming, but a time constraint wasn't part of the question. A mass email to all of the might net some responses straight off the bat, reducing the number of calls you need to make.
It's worth noting as well that a lot of companies are quite lax when it comes to this stuff. I've been able to get past data protection questions with answers that anyone else purporting to be me could easily give. Just today, as part of my job, I needed to contact a utility company and I failed a data protection question. I was able to get past it by simply sending them a blank email from my work email address, and the fact that the name on the account was a partial match to the domain name of my email address, was sufficient. It is trivially easy to fake such an email.