In order to use Mace’s ability, the cost needs to be paid in full. That means you have to both stand a character and successfully remove it from the challenge. You can only pay costs with cards you control.
On the other hand, the character you choose as the target of Mace’s ability can be controlled by any player. You can choose a character that is already standing (Ser Jaime Lannister (Core)), or a character that cannot really be removed from the challenge (for instance, if Walder Frey is attacking), as long as the character can be affected by at least one aspect of the effect.