A wig is a head covering made from human hair, animal hair, or synthetic fiber. A wig may be used as a less intrusive and less expensive alternative to medical therapies for restoring hair or for a religious reason.
Synthetic wigs are allowed whether made out of human hair or out of any other material. The point is when the wigs are planted in the skin, such wigs prevent water from reaching to the skin and makes a problem in the Wudhu. If wigs are removed before performing Wudhu then there is no problem at all. Wigs can be even worn after performing Wudhu while performing Salah Namaz if the wigs are Tahir. In case Wigs are made of human hair from Najis source like Mushriks then Salah with such Najis hair will be invalid.
Wearing wig is allowed in Islam, but if a woman wears it, she must cover it as a part of her Hijab obligation. Some Jewish women wear wig and consider it a full cover of the hair but in Islam wearing wig does not mean any change in Hijab obligation.
Those who need to wear wigs should not hide it when they propose or are been proposed for marriage as marriage must be built on true facts about both with out hiding such things.