The current version of the Skype app isn't compatible with an iPhone 5. See the following from Skype for iPhone on the App Store
Compatibility: Requires iOS 9.0 or later. Compatible with iPhone 5s, iPhone 6, iPhone 6 Plus, iPhone 6s, iPhone 6s Plus, iPhone SE, iPhone 7, iPhone 7 Plus, iPad Air, iPad Air Wi-Fi + Cellular, iPad mini 2, iPad mini 2 Wi-Fi + Cellular, iPad Air 2, iPad Air 2 Wi-Fi + Cellular, iPad mini 3, iPad mini 3 Wi-Fi + Cellular, iPad mini 4, iPad mini 4 Wi-Fi + Cellular, 12.9-inch iPad Pro, 12.9-inch iPad Pro Wi-Fi + Cellular, 9.7-inch iPad Pro, 9.7-inch iPad Pro Wi-Fi + Cellular, iPad (5th generation), iPad Wi-Fi + Cellular (5th generation), 12.9-inch iPad Pro (2nd generation), 12.9-inch iPad Pro Wi‑Fi + Cellular (2nd generation), 10.5-inch iPad Pro, 10.5-inch iPad Pro Wi‑Fi + Cellular, and iPod touch (6th generation).