Slicha's closest equivalent would be Pardon (including how it's used in french). it can be used as a casual apology or as "excuse me?"
I don't know if it counts as a "formal" way, but there's the more deliberate form:
Ani Mizta'er / Ani Mizta'eret ("I'm sorry", m/f) אני מצטער\מצטערת
"mizta'er/et" alone (without "ani") is more casual.
And if you really messed things up, you may want to add "Ani Mitnazel/et" אני מתנצל\ת - I apologize :)