The MT for Judges 6:37 is:
הִנֵּה אָנֹכִי מַצִּיג אֶת גִּזַּת הַצֶּמֶר בַּגֹּרֶן אִם טַל יִהְיֶה עַל הַגִּזָּה לְבַדָּהּ וְעַל כָּל הָאָרֶץ חֹרֶב וְיָדַעְתִּי כִּי תוֹשִׁיעַ בְּיָדִי אֶת יִשְׂרָאֵל כַּאֲשֶׁר דִּבַּרְתָּ
The operative word is גִּזַּת, jizat, meaning "a shearing of", an intertwined mass of wool sheared from a sheep. See how the same word is used in Deuteronomy 18:4:
רֵאשִׁית דְּגָנְךָ תִּירֹשְׁךָ וְיִצְהָרֶךָ וְרֵאשִׁית גֵּז צֹאנְךָ תִּתֶּן לּוֹ
(NIV) You are to give them the firstfruits of your grain, new wine and olive oil, and the first wool from the shearing of your sheep
This same word in various conjugations, gazaz, gozez, gozazim or ligzoz, appears in:
- Genesis 31:19
- Genesis 38:12
- I Samuel 25:2, 4, 7, 11
- II Samuel 13:23, 24
- Isaiah 53:7
- I Chronicles 2:46
OTOH, a sheepskin or other animal skin with or without the hair is simply a "skin" or "hide", עור, as in Leviticus 7:8
וְהַכֹּהֵן הַמַּקְרִיב אֶת-עֹלַת אִישׁ עוֹר הָעֹלָה אֲשֶׁר הִקְרִיב לַכֹּהֵן לוֹ יִהְיֶה
(NIV) The priest who offers a burnt offering for anyone may keep its hide for himself