Speaking during a live link-up with a primary school in London, he also said he was "immensely proud" to be wearing the Union Flag.
Major Peake added that he was now into the final phase of training, in which he was focusing on the essentials.
He said astronaut training had been rigorous and involved retaining immense quantities of information.
Tim Peake is the first Briton to be selected as an astronaut for the European Space Agency, through a route backed by the UK government.
He was in Houston, Texas, when he spoke live to schoolchildren at Queen's Park Primary School in West London. The event was organised as part of a UK Space Agency programme called Esero, which is designed to use space to help enhance science education.
"I'm immensely proud to be British and to be serving in the European Space Agency," he told BBC News.
"It's always very apparent to me, wherever I'm travelling, whether it be Houston, here, or Russia, that you're wearing the flag and you're representing your country."
Major Peake, who was a British Army helicopter pilot before his selection for the astronaut corps, is set to fly to the International Space Station in December on a Russian Soyuz rocket.
He will be accompanied on the launch from Baikonur Cosmodrome by the American astronaut Timothy Kopra and the Russian Yuri Malenchenko.
The Briton said the training was now "ramping up" as the 15 December launch date approached.
"The training is a long process and we've covered such a huge variety of subjects. One of the main challenges is actually being able to retain all of that information over a two-and-a-half-year period," he said.
"I'm in the final six months before launch and we're just focusing on the essentials of the Soyuz spacecraft, emergency training... and scientific payloads that are going to be on the space station."
Pupils who listened to Major Peake talk, and got the opportunity to ask him questions, seemed to be impressed by the experience.
Abdullah Hashim, who is 10, said he was "an extraordinary man" and an "inspiration". Salma Sarumi, 8, said it was "amazing" to have the live link-up at her school.
Fellow students Cherise Panzu, Mario Albu and Susu El-Abbas also seemed to have been inspired by the event. Cherise called it a "once in a lifetime opportunity".
Major Peake believes the flight will bring significant benefits to the country and British industry.
"This is a step forward in that it's the UK government supporting human spaceflight," he explained.
"What that means is that UK industry, UK education and the UK scientific community can now embrace human spaceflight. That means taking part in scientific research that's being conducted not only on the International Space Station, but within the other European Space Agency human spaceflight programmes."
The UK's first official astronaut, Tim Peake, says he is "well prepared" as his December launch date approaches.