I am being paid interest into my bank account and not really sure who to give it to.
Is it simple enough for me to donate it to the mosque or does it have to be donated to somewhere specific?
Some scholars feel that purifying ones own wealth by donating such money directly to the masjid is a bad idea, since the masjid is a house of God and should be built and maintained using only pure funds. Similarly, it shouldn't be used to print Qur'ans.
Rather it should be given to the poor and the needy, or any other charitable purpose. This is, I believe, the Hanafi position on the matter.
There's a difference of opinion about this, and there are two schools of thought.
But first, have you considered switching to an interest-free account? Several banks offer these, some upon request (even if it's not advertised).
The two opinions are:
Don't touch it. It's riba money, and you should leave it where it is. When you close your account, close it with all the riba interest intact.
Get rid of it. It's not really your money. You can give it away, but don't expect any reward for it, because the money is not pure. As rasulullah said in one hadith:
"Allah is pure (tayyeb) and only accepts pure (tayyeb)." (Jami' at-Tirmidhi).
And Allah knows best.