Of course you can, it is called proof of stake (for instance).  Some call it "check points" (which is signature of the dev).  Proof of stake has a known problem which is "nothing at stake", but ONLY because one combines proof of stake also with block rewards.  Proof of stake as a non-rewarded consensus mechanism doesn't suffer from that.  

Bitcoin could not really start with *rewarded* PoS, because then Satoshi was the only one who could "stake" in the beginning, so he would be the only one receiving new coins, and the only people obtaining bitcoin would be people getting them directly from Satoshi, which would look too much like Satoshi printing an infinite amount of money that he would then sell.
However, Satoshi could have had the idea of using PoW for coin CREATION, but keeping the consensus mechanism with PoS, non-rewarded.  At that time, it probably seemed a good idea to reward people doing the consensus signature thing ; but it turned bitcoin in a "mining industry" where the users are now "external customers".  If the consensus mechanism had been based upon staking signatures (with no reward) then "miners" would only be people making new coins, but would have had nothing to say about the protocol, the block size, or whatever, which would have remained in the hands of the bitcoin owners (the stakers).
It seemed a good idea to do everything in one go, but it turns out it probably wasn't.