After I win a showdown, can I use Bone Skewer to initiate another showdown?

Yes, you can use Bone Skewer to initiate another showdown, provided you satisfy the game's timing and state requirements.

Timing and Execution

According to the rules for Showdowns (Rule 342, Rule 344), a showdown begins when a battlefield is contested. Because your previous showdown has concluded, you are back in a "Neutral Open State."

1. Playing the Spell: Since Bone Skewer is a spell, you can play it during your Main Phase while the game state is open.
2. Creating the Contested Status: Bone Skewer instructs you to choose a battlefield and have your opponent play a unit from their hand to that location. If that battlefield was previously uncontrolled or had no units from your opponent, placing their unit there makes the battlefield contested between you and your opponent.
3. Initiating the Showdown: Per Rule 344.1, if control of a battlefield becomes contested between two players, a showdown is opened. Since you are placing an opponent's unit at a battlefield where you presumably have your own units (or will move them), this will trigger the conditions to start a new showdown.

Important Considerations

• Combat vs. Non-Combat: If your unit is already present at the battlefield where you play the opponent's unit, the showdown will open as a Combat Showdown because units controlled by different players are present (Rule 459.2).
• Spell Timing: You can play Bone Skewer as an action because it is a spell and the game state is open. However, if you are currently in a showdown, you would not be able to play Bone Skewer because it lacks the [ACTION] or [REACTION] keyword (refer to RiftJudge FAQ #9671). Since you stated you have already won the previous showdown, you are back in the main phase and free to play it.
• Rule 465: Keep in mind that per Rule 465, you can only score from a battlefield once per turn. Winning a second showdown at the same location will not grant you a second point for that battlefield.

This is my interpretation based on the standard rules for initiating showdowns and playing spells. No official FAQ exists for this specific interaction.