Cancel other person plane ticket that I bought

Did you try to call your airline? IMHO you should not mention about the ex.

But many airplane tickets cannot be really cancelled, reimbursement is also seldom on usual fares, but eventually you can change the person name.

On the other hand, if you gave the ticket to your ex, she owns the ticket. You did the error. Past is past.

Any ticket bought can be cancelled by the purchaser. Not all tickets are refundable though, so you may not get your money back. Many airlines (but not all) allow name changes on a ticket, so you could call and see if you can have it changed to someone else. There is usually a fee associated with this; expect to pay $100-200 USD for standard carriers.

The Airline is going to be your easiest point of contact. No need to give them any reasons. Call and ask to cancel the ticket in question. If it is refundable, they will let you know and ask for information to process the refund. Even if it is refundable, you may not get all money back but it will not be usable by anyone.

It is possible to have a travel agent do that for you. Mostly, if you purchased the ticket through them. Again, they do not need reasons, just to know that you are cancelling that ticket.

You did not specify the itinerary but in some cases the ticket cannot be used without the presence of the purchaser who must show the card used to buy the ticket at check in. Personally, this has never happened to me, but it is apparently a common practice of some Asian and Middle Eastern airlines.