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The men's team set an Olympic record with a shock victory against world champions New Zealand.
"There's no better way to win: beating the world champions; setting the Olympic record and getting the gold medals," said the Scot.
Sir Chris Hoy praised his "phenomenal" performance when it mattered most.
It was Britain's fourth gold at Rio 2016, taking their overall medal tally to 16 as Skinner reflected on the first gold for a Scottish competitor.
"We felt we were getting there but always felt we were on the back foot," he said.
"In the last few World Championships we haven't managed to string it together and, to be honest, if we made the [Olympic] final then we'd have been very happy with that.
"So, for it all to come together as well as it has done is just amazing. I looked at the bookies the other day and it was like 14-1 for us to win! To set two Olympic records is fantastic - so, over the moon!"
Skinner said the legacy of the Rio Games should be improving facilities in order to produce more talent in Scotland.
"Scotland used to have one velodrome which was outdoors and was the equivalent of all the Scottish swimmers training in one outdoor pool," he added.
"It's great to see investment in the facilities and it'd be great if Edinburgh City Council could build us an indoor velodrome but I know they're committed to an outdoor one.
"It [cycling] seems to be something Scotland gets relative success from; with more investment in facilities then the names will come through."
Sir Chris Hoy, six-time Olympic cycling champion:
"I was looking at a picture earlier on that someone had posted on Twitter that had been taken about 12 or 13 years ago from Meadowbank. It was me holding him [Callum Skinner] up for a standing start; and he had little stick legs and he was only about 12 at that point.
"It just brought it home how far he's come from being that young lad training at Meadowbank, where I started, and so many of the Scottish riders started there.
"The way he dealt with that pressure tonight, he knew it was all on his shoulders because he had the fastest man in the world going off the first lap and the fastest man in the world off the second lap.
"It was all on his shoulders and he was so calm - just phenomenal. And to finish with an Olympic record on top of the gold medal: amazing.
"For people who don't watch track cycling within that four-year period and then watch it at the Olympic Games, they may be forgiven for thinking this was just another medal for the Team GB who always seem to win gold medals.
"But this was one of the least predicted ones; they haven't won a medal at world level in the past six years.
"There wasn't a massive expectation but in the last six months in training they've really stepped it up and they just came together at the same time.
"That's what the Olympics is all about: it's about performing on the day that counts."
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Mr Jammeh told state TV the proclamation was in line with Gambia's "religious identity and values".
He added that no dress code would be imposed and citizens of other faiths would be allowed to practise freely.
Some 90% of Gambians are Muslim. The former British colony's economy relies heavily on tourism.
However, relations with the West have soured recently.
The European Union temporarily withheld aid money to Gambia last year over its poor human rights record.
Mr Jammeh has been president of the tiny West African country for 21 years.
"As Muslims are the majority in the country, the Gambia cannot afford to continue the colonial legacy," Mr Jammeh told state TV, explaining his decision to proclaim an Islamic republic.
Other Islamic republics include Iran, Pakistan and - in Africa, Mauritania.
Mr Jammeh withdrew Gambia from the Commonwealth in 2013, describing the organisation as neo-colonial.
In 2007, he claimed to have found a herbal cure for Aids.
Callum Skinner is "over the moon" after securing gold for Team GB in the men's team sprint - and joked about their 14-1 odds with the bookmakers.
[NEXT_CONCEPT]
Gambian President Yahya Jammeh has declared his Muslim-majority country an Islamic republic, saying the move marks a break with the colonial past.