Making a door from 2x4 is relatively simple..........you say your 4x4 posts are 36 inches apart and I'll assume they are 72 inches tall so your door opening in 36x72.............make your gate 35 x 71 with simple butt jointed 2x4s...........that will give you 1/2 inch clearance on all 4 sides of run door.
If you have a miter saw.............cut 2x4 scraps with 45 degree angles on both ends...........quick and easy corner braces...........or use gussets made from plywood scrap............google plywood gusset..........or buy steel L-angles braces at hardware store
when hanging door use shims to insure that you hAVE 1/2 inch clearance at bottom of door,at top of door and between door edge and 4x4......hold in place with clamps while you screw your hinges in place
good luck