When Bone Skewer is played from hidden, can I choose another battlefield other than the battlefield it was hidden?

No — if you play Bone Skewer from face down, the battlefield you choose must be the one it was hidden at and played from.

Bone Skewer's instruction makes a unit get played, so Rule 811.1.d.5 applies: a hidden spell (or hidden permanent's play effect) that causes you to play a unit forces you to play it at the battlefield the spell was played from. The "choose a battlefield" line collapses to your own hidden battlefield — you can't reach across to force a unit onto a different one. The play is still legal because your own battlefield is always a valid choice, so the spell resolves normally, just locked to that location (FAQ #13108).

If you play Bone Skewer from hand for its cost at Action timing instead, you're free to pick any battlefield.