The angel of the Lord first appears as the angel of the Lord at Genesis 16:7. And he says in the following verses that he will multiply Hagar's descendants at verse 10. At verse 13 Hagar says, "Thou art a God who sees; Have I even remained alive after seeing Him."
At Genesis 17:1-2 you have the Lord appearing to Abram and said to him, "I am God Almighty; Walk before Me, and be blameless. verse 2, And I will establish My covenant between Me and you, And I will multiply your exceedingly."
Going to Genesis 22:1 it says, "Now it came about after these things, that God tested Abraham etc." At verse 11, "But the angel of the Lord called to him/Abraham from heaven, AND SAID, Abraham, Abraham! etc." verse 12, "Do not stretch out your hand against the lad, and do nothing to him, for now I KNOW that you fear God, since you have not WITHHELD your son, your only son, FROM ME."
Verse 15, "Then the angel of the Lord called to Abraham a second time from heaven, verse 16, AND SAID, "By Myself I have sworn, declares the Lord, because you have done this thing, and have not withheld your son, your only son, verse 17, "indeed I will greatly multiply your seed as the starts of the heavens, and as the sand which is on the seashore; and your seed shall possess the gate of their enemies." Verse 18, "And in your seed all the nations of the earth shall be blessed because YOU HAVE OBEYED MY VOICE."
So what we have here is the angel of the Lord at Genesis 16 who multiplied Hagar's descendants and was identified as God is the same being who made the covenant to Abram at Genesis 17:1-2 who is identified as God. It is also the same being (the angel of the Lord) at Genesis 22:16-18.
All of this is backed up by the writer of Hebrews at Hebrews 6:13-14, "For when God made the promise to Abraham, since He could swear by no one greater, HE SWORD BY HIMSELF, Verse 14, "saying, I will surely bless you, and I will surely multiply you." At Luke 1:73, "The oath which He swore to Abraham our father."
A couple of points of interest. The angel of the Lord is speaking in the "first person" here. This is confirmed at (Genesis 16:10, Genesis 21:18, Genesis 31:13). Also the angel of the Lord appears at, Genesis 16:7-14, Genesis 22:11-16, Exodus 3:2-4, (the burning bush), Numbers 22:22-38, Judges 2:1-3 and at other places.
Also note, that the angel of the Lord never appears in the New Testament, although he is mentioned at Acts 7. Another important point that people miss is there is a difference between the words, "a or an" as opposed to the word "the." For example, some will say the angel of the Lord appeared to Joseph at Matthew 1:20 and say "the angle of the Lord appeared etc. despite the fact that the word used is "an" angel of the Lord appeared.
The grammatical function of "an" or "a" is to connote a thing not previously noted or recognized. In other words, it's just that, an angel/messenger. The word "the" connotes a thing PREVIOUSLY noted or recognized, i.e "THE angel of the Lord. Finally, I'm convinced by the evidence that the angel of the Lord is the preincarnate Jesus Christ.