"Ik geef jou toestemming om in de lucht te springen."
Translation:I give you permission to jump in the air.
That would be perfectly fine in this case. Although in a sentense like this, 'jou' would be used to more specifically point somebody out when there are more people around. '... je' would be used when there really is nobody else it could apply to.
There's no article in the English translation and in Dutch it's not necessary to put "de" in front of "toestemming." I give you permission to jump off that bridge =Ik geef jou toestemming om van die brug te springen.
Why is om te here? I don't think there can be a subclause, or an 'in order to' meaning...