Cross-validation strategy I have a regression problem and I am in doubt about how I can calculate RMSE in my life-cycle.

I deal with time-series and for every prediction, I want to look N points in the future. It is apparent how to calculate RMSE for a single iteration. My question is how to calculate RMSE for N predictions of N points to get a meaningful prediction performance metric.

I guess, I can average RMSE of all iterations though as I said I am not sure at all if this would reflect actual performance.
The natural choice would be the total squared error across the N predicted values, averaged across all examples. This is the simple extension of mean squared error from the univariate case. If you're using multivariate linear regression, this is in fact what you want to optimize in order to get the maximum likelihood estimate of the parameters as well.