Intention is not required for wiping over the socks to be permissible as long as they are worn in a state of purity.

You can do wudu with socks on if the socks have been put on when you are in a state of purity and they are pure, and the wiping may be done within the time specified by Shari`ah.

Wiping over the socks is limited to one day and one night for the non-traveller, and three days and nights for the traveller.When the period ends, it is not permissible to wipe over the socks until you take them off, then put them on after purifying yourself fully with water. You can overcome the difficulty of washing your feet by sitting on a chair, or pouring water over the feet without bending down to wash them.

What must be done in Wudu is washing the feet; it is not sufficient to wipe them. But the Sunnah shows that one may wipe over the feet only when wearing leather slippers or socks, subject to the conditions that are well known in the Sunnah.

The time for wiping over socks may expire but your Wudu remains valid as long as you do not perform any act that breaks it.

There is no evidence to indicate that taking off leather slippers or socks that have been wiped over invalidates Wudu. If you wish to wipe over the socks again after breaking Wudu, you must perform a complete wudu, including washing your feet, before putting the socks back on.