"It seems to be a common thing with Scottish rugby - you should have won that game," Horne told BBC Scotland's Rugby Podcast.
"We've just got to go out and actually just get it done. We're conscious of that, especially as a senior player group; that when it gets to these big games that are swung on tiny moments, we're not getting it done.
"It's something that's happening at Glasgow - Harlequins game [in the Champions Cup], [Challenge Cup] final last year, Munster game in the [United Rugby Championship] quarter last year and then it's kind of a similar thing with Scotland.
"We know we've got the players and in games we've got the opportunities to do things and we've not been able to. So, it's something that I think at the end of the day we're just going to have to go and do.
"We can't talk about how close we are, it's just actually acting on that in the game. We're so desperate to get it done but until that happens it's just something that's so frustrating.
"Once it happens once, hopefully we can kind of breed that winning culture. And you've seen the likes of Leinster, Ireland - they're used to winning all these tight games. They just know how to get the job done. It's something we can hopefully improve on."
Glasgow head coach Franco Smith said recently Scottish players get "too nervous" for the biggest games and become too focussed on the final result rather than the detail around performance.
Horne agrees that a desperation to win has proved counterproductive.
"Franco talks about that a lot," Horne said. "Because we're so desperate, maybe you'll do something extra that you don't need to do or you'll go away from something in the game plan or you'll try this or try that.
"Whereas the best teams in the world execute their plan, they stick to what they're good at and they get it done.
"Personally, it's something I'm bad for. I'm so desperate to give my all and give my best, I'll probably end up trying to do too much and it'll negatively impact the team.
"We've got the players, we've got the 23, we've got the squad. If we just do our job on the day, we should have enough to be hopefully going to win."