Ms Swinson admitted that most men in her position would "run for leader like a shot", but said that that did not in itself make it the right decision.
The MP, who won back her East Dunbartonshire seat from the SNP at the General Election, said that she had been "overwhelmed by so many lovely messages from people I know, and from many members I have not yet met, encouraging me to stand for leader. I am touched and flattered that you look to me and I am determined to play a key role in our party's leadership."
But she added that the role of leader of a political party was a unique and "all-encompassing job" and said that it should not be "done simply to achieve status, to make a point, or to please others".
Ms Swinson, who is writing a book on equality, added: "Feminist that I am, I have of course wondered what a bloke in my position would do. It's obvious. Most blokes in my shoes would run for leader like a shot. It's true that my many years of encouraging women to have the confidence to go for that exciting new role have taught me that women often don't go for things when they should.
"But just as often I have observed men going for the promotion when they shouldn't. Just because a man would do it, doesn't make it the right thing to do. I have consistently fought against stereotypes and structures that impose a choice on someone, rather than allowing them to make up their own mind."
In a highly unusual development, politicians from other parties urged Ms Swinson to change her mind.
Former Labour deputy leader Harriet Harman tweeted: "Oh no, Jo! Go for top job. And the men can aspire to be your deputy!!"
Mr Farron, a committed Christian who was repeatedly asked during the election campaign if he believed gay sex was a sin, resigned last week saying his religion had made him a "subject of suspicion".
His resignation means that the party is facing its second hunt for a new leader in two years.
Among those expected to stand are former Business Secretary Vince Cable, former health minister Norman Lamb and possibly former Energy Secretary Ed Davey.