Hooker Dewi Lake, the Cherry and Whites' new captain, scored two tries, while his Wales teammate Jac Morgan and South African lock Jean Kleyn also starred at Sandy Park in a convincing defeat of Exeter, last season's losing finalists.
"The result is great, but what we saw in character, particularly in some of those defensive sets from the boys, is a good snapshot of what we have asked them to go after this year," said Skivington.
He added: "I thought Dewi did well, Jac had a lot of good moments in the game in the nitty-gritty, and Jean really showed his presence in the first half, but I thought we did well across the board."
Exeter boss Rob Baxter refused to blame the loss on the illness that ravaged his squad in the build-up
"If it had been a real battle and we had tried really hard with most of the stuff we said we were going to do, and Gloucester came through on top, then I would have said 'fair play, we have done pretty well considering the issues we had', but I think we were a good notch below that and I thought we were quite poor," he said.
Saturday's other Prem match saw Wales star Louis Rees-Zammit score two of Bristol's five tries in a 34-24 win at Sale.
On Friday, champions Northampton ran in nine tries as they launched their title defence with a 59-26 win at home to Newcastle.
Bath, meanwhile, beat Harlequins 33-17 at the Stoop but victory was overshadowed by a biceps injury to Sam Underhill that could threaten the England back-row forward's involvement in next month's Nations Championship internationals.