How to check machine learning accuracy without cross validation
It is not necessary to do 75|25 split of your data all the time. 75
|25 is kind of old school now. It greatly depends on the amount of data that you have. For example, if you have 1 billion sentences for training a language model, it is not necessary to reserve 25% for testing. 
Also, I second the previous answer of trying K-fold cross-validation. As a side note, you could consider looking at the other metrics like precision and recall as well.
I have training sample X_train, and Y_train to train and X_estimated.
I got task to make my classificator learn as accurate as it can, and then predict vector of results over X_estimated to get close results to Y_estimated (which i have now, and I have to be as much precise as it can). If I split my training data to like 75/25 to train and test it, I can get accuracy using sklearn.metrics.accuracy_score and confusion matrix. But I am losing that 25% of samples, that would make my predictions more accurate. 
Is there any way, I could learn by using 100% of the data, and still be able to see accuracy score (or percentage), so I can predict it many times, and save best (%) result?
I am using random forest with 500 estimators, and usually get like 90% accuracy. I want to save best prediction vector as possible for my task, without splitting any data (not wasting anything), but still be able to calculate accuracy (so I can save best prediction vector) from multiple attempts (random forest always shows different results)
Thank you