Adkins was dismissed after a terrible start to the season and leaves Charlton in the bottom three of League One, with just two wins from 13 matches.
Jackson, 39, spent eight years as a Charlton player and was assistant to former boss Lee Bowyer as well as Adkins, having begun coaching at The Valley in 2017.
Asked if he wanted the full-time job, Jackson told Standard Sport: “Definitely. Obviously it’s long been an ambition of mine to manage and certainly to manage this club. I started my coaching career here, had a long playing career here, so I’ve got an affinity with the club. I’d love the opportunity to become the manager.
“I’ve got confidence in myself. I certainly think I’m in a much better position now than maybe I have been in previous times, when I’ve perhaps flirted with [the job]. I feel I’m in a good place, I’ve been coaching for a good number of years, I’ve got experience. I played under a lot of different managers and styles and now I’ve had the opportunity to work with a few as well. There’s always more to learn but I definitely feel I’ll be capable of having an impact.
“I spoke to Thomas in March about taking charge [as caretaker after Bowyer’s resignation]. I did make it clear to him at that stage that it was an ambition of mine to be manager of this football club and I’d love to be considered. Obviously at that stage they decided to go a different way. You always think you’re ready but today I think I’m in a better position in terms of my coaching and experience than I was even six months ago.
“I’ve got my own ideas. Nigel is an experienced manager, he’s got his own. He was great with me and gave me a lot of leeway, but ultimately he was the manager and called the shots.
“Likewise, now I’m in charge and hopefully that’s for a long time but while I am, I’ll be looking to put my stamp on things and I’ve got ideas of what we need to change. We need to change things because what we’re doing isn’t working, clearly. That’s my job, my responsibility, to put my own framework on it and start getting results.”
Ever the optimist, Sandgaard says Charlton can still reach the play-offs this season but Jackson insists he is only looking at short-term results to lift the club away from relegation danger.
“My immediate focus has got to be on the short-term,” he said. “We find ourselves in the relegation zone so we have to put a few results together and get out of that situation. Then maybe we can start looking up.”
Meanwhile, Sandgaard has hinted he is in talks with legendary former Charlton boss Alan Curbishley about involvement on the football side of the club.
Curbishley managed Charlton from 1991 to 2006, and has been working for the club’s in-house television channel.
Asked by BBC Radio London if he could look to bring in some experienced heads who know the club, Sandgaard replied: “That was asked very generically but obviously I know what you mean.
“And yes, I am already talking frequently to the people you are talking about. I am certainly not ruling out bringing some of those people at some level to be more involved with the club. That is yet to be seen. Let’s see how those talks go.”