Linux

Yes, it can be done. Not to the point of deleting your key (that makes no sense - you need the key), but ssh-agent is what you want. Add it to your shell config and it will only ask to be unlocked once per however often you define.

I have this function defined and called:

function ssh-agent-setup() {
    # SSH agent
    pid_file="$HOME/.ssh/ssh-agent.pid"
    SSH_AUTH_SOCK="$HOME/.ssh/ssh-agent.sock"
    if [ -z "$SSH_AGENT_PID" ]
    then
      # no PID exported, try to get it from pidfile
      SSH_AGENT_PID=$(cat "$pid_file")
    fi

    if ! kill -0 "$SSH_AGENT_PID" &> /dev/null
    then
      # the agent is not running, start it
      rm "$SSH_AUTH_SOCK" &> /dev/null
      >&2 echo "Starting SSH agent, since it's not running; this can take a moment"
      eval "$(ssh-agent -s -a "$SSH_AUTH_SOCK")"
      echo "$SSH_AGENT_PID" > "$pid_file"

      >&2 echo "Started ssh-agent with '$SSH_AUTH_SOCK'"
    fi
    export SSH_AGENT_PID
    export SSH_AUTH_SOCK
}

ssh-agent-setup

This way it stores the unlocked key in memory until the end of the session.