The Word Reference dictionary and Wiktionary say that the correct pronunciation of the verb "excuser" is [ɛkskyze] with a open E at the beginning (as in hiEr), but I hear a closed E (/e/) in the beginning of the word in the available audios in the two webpages. Am I listening right?
Excuser can have several pronunciations:
The careful/formal one is /ɛkskyze/.
Due to the difficulty of the sequence /ksk/, a more common pronunciation is dropping the first /k/ leading to /ɛskyze/.
The first vowel might also be partially or totally dropped in relaxed French, especially in conjugated forms like excusez-moi, leading to /skuze mwa/.
Using /e/ like in the first linked page (/ekskyze/) doesn't prevent understanding and would likely be unnoticed by most native speakers.
I also hear an [e] on the French recording on the Word Reference dictionary. To my hears, it sounds quite artificial and not representative of the common pronunciation with [ɛ]. Note that on the Canadian audio, you can clearly hear an [ɛ].
On the Wiktionary entry, on the contrary, I hear a clear [ɛ]. It is a pity that the audio is for s'excuser and not excuser alone.