The following source is already a bit dated. I'm not a native speaker and don't have a strong opinion on the matter.
The American Heritage® Book of English Usage A Practical and Authoritative Guide to Contemporary English. 1996. 3. Word Choice: New Uses, Common Confusion, and Constraints § 23. all right / alright:
"Is it all right to use alright? Despite the appearance of alright in the works of such well-known writers as Flannery O’Connor, Langston Hughes, and James Joyce, the merger of all and right has never been accepted as standard. This is peculiar, since similar fusions like already and altogether have never raised any objections. The difference may lie in the fact that already and altogether became single words back in the Middle Ages, whereas alright (at least in its current meaning) has only been around for a little over a century and was called out by language critics as a misspelling. You might think a century would be plenty of time for such an unimposing spelling to gain acceptance as a standard variant, and you will undoubtedly come across alright in magazine and newspaper articles. But if you decide to use alright, especially in formal writing, you run the risk that some of your readers will view it as an error, while others may think you are willfully breaking convention."