I'm going to court a little controversy and say that you should not ask for another interview.
As an interviewer, I have to assume that you are performing at your best. I can't think "Well, that was a pretty poor interview, but I bet Parth is better when not under pressure."
In your day to day work, I need to know that you will be able to cope with the pressure of deadlines, urgent client meetings, things breaking. If you can't cope with a friendly chat over Skype - am I going to be happy putting you in from of my boss?
That said, yes, we know everyone gets nervous in interviews. We're not expecting 100% perfection. You probably didn't perform as badly as you thought.
The interviewer is the one who decides which candidate gets a second interview - not you. Do not write saying "Gosh! I was pretty crappy, eh? Can you give me a second chance? PLEASE!"
Instead, say something like...
Thanks for taking the time to interview me last week. I found the whole process really interesting. Based on the questions I asked, I would love to work for XYZ Corp. I really think my experience with ABC could be a great asset to your project - and I'm keen to learn more about your 123 processes.
Please do let me know if you need any more information from me.
That way, you are showing that you actually want the job and get to reiterate your strengths.